Frankly, I'm not sure whether the distinction between "worlds" and "experiences" is more useful or more harmful. There is definitely something that rings true about your post but people have been misinterpreting all that in a very silly ways for decades and it seems that you are ready to go in the same direction, considering your mentioning of anthropics.

Mathematically, there are mutually exclusive outcomes which can be combined into events. It doesn't matter whether these outcomes represent worlds or possible experiences in one world or whatever else - as long as they are truly mutually exclusive we can lawfully use probability theory. If they are not, then saying the phrase "1st person perspective" doesn't suddenly allow us to use it.

How do we give $A$ the intuitive meaning of "my sensor sees red"?

We don't, unless we can formally specify what "my" means. Until then we are talking about the truth of statements "Red light was observed" and "Red light was not observed". And if our mathematical model doesn't track any other information, then for the sake of this mathematical model all the robots that observe red are the same entity. The whole point of math is that it's true not just for one specific person but for everyone satisfying the conditions. That's what makes it useful.

Suppose I'm observing a dice roll and I wonder what is the probability that the result will be "4". The mathematical model that tells me that it's 1/6 also tells the same to you, or any other person. It tells the same fact about any other roll of any other dice with similar relevant properties. 

From this, we get a nice potential explanation to the Sleeping Beauty paradox: 1/2 is the 0P-probability, and 1/3 is the 1P-probability. This could also explain why both intuitions are so strong.

I was worried that you would go there. There is only one lawful way to define probability in Sleeping Beauty problem. The crux of disagreement between thirders and halfers is whether this awakening should be modeled as random awakening between three equiprobable mutually exclusive outcomes: Heads&Monday, Tails&Monday and Tails&Tuesday. And there is one correct answer to it - no it should not. We can formally prove that if Tails&Monday awakening is always followed by Tails&Tuesday awakening, then they are not mutually exclusive.

I disagree with the whole distinction. "My sensor" is indexical. By saying it from my own mouth, I effectively point at myself: "I'm talking about this guy here." Also, your possible worlds are not connected to each other. "I" "could" stand up right now because the version of me that stands up would share a common past with the other versions, namely my present and my past, but you haven't provided a common past between your possible worlds, so there is no question of the robots from different ones sharing an identity. As for picking out one robot from the others in the same world, you might equally wonder how we can refer to this or that inanimate object without confusing it with indistinguishable ones. The answer is the same: indexicals. We distinguish them by pointing at them. There is nothing special about the question of talking about ourselves. As for the sleeping beauty problem, it's a matter of simple conditional probability: "when you wake up, what's the probability?" 1/3. If you had asked, "when you wake up, what's the probability of having woken up?" The answer would be one, even though you might never wake up. The fact that you are a (or "the") conscious observer doesn't matter. It works the same from the third person. If, instead of waking someone up, the scientist had planned to throw a rock on the ground, and the question were, "when the rock is thrown on the ground, what's the probability of the coin having come up heads?" It would all work out the same. The answer would still be 1/3.

This approach implies there are two possible types of meanings: Sets of possible worlds and sets of possible experiences. A set of possible worlds would constitute truth conditions for "objective" statements about the external world, while a set of experience conditions would constitute verification conditions for subjective statements, i.e. statements about the current internal states of the agent.

However, it seems like a statement can mix both external or internal affairs, which would make the 0P/1P distinction problematic. Consider Wei Dai's example of "I will see red". It expresses a relation between the current agent ("I") and its hypothetical "future self". "I" is presumably an internal object, since the agent can refer to itself or its experiences independently of how the external world turns out to be constituted. The future agent, however, is an external object relative to the current agent which makes the statement. It must be external because its existence is uncertain to the present agent. Same for the future experience of red.

Then the statement "I will see red" could be formalized as follows, where $i$ ("I"/"me"/"myself") is an individual constant which refers to the present agent:

$$\exists x\left(WillBecome\right(i,x)\wedge \exists y(Red\left(y\right)\wedge Sees(x,y)\left)\right).$$

Somewhat less formally: "There is an x such that I will become x and there is a experience of red y such that x sees y."

(The quantifier is assumed to range over all objects irrespective of when they exist in time.)

If there is a future object $x$ and a future experience $y$ that make this statement true, they would be external to the present agent who is making the statement. But $i$ is internal to the present agent, as it is the (present) agent itself. (Consider Descartes demon currently misleading you about the existence of the external world. Even in that case you could be certain that you exist. So you aren't something external.)

So Wei's statement seems partially internal and partially external, and it is not clear whether its meaning can be either a set of experiences or a set of possible worlds on the 0P/1P theory. So it seems a unified account is needed.

Here is an alternative theory.

Assume the meaning of a statement is instead a set of experience/degree-of-confirmation pairs. That is, two statements have the same meaning if they get confirmed/disconfirmed to the same degree for all possible experiences that E. So statement A has the same meaning as a statement B iff:

$$\forall E\left(P\right(A\mid E)=P(B\mid E\left)\right),$$

where $P(\_\mid \_)$ is a probability function describing conditional beliefs. (See Yudkowsky's anticipated experiences. Or Rudolf Carnap's liberal verificationism, which considers degrees of confirmation instead of Wittgenstein's strict verification.)

Now this arguably makes sense for statements about external affairs: If I make two statements, and I would regard them to be confirmed or disconfirmed to the same degree by the same evidence, that would plausibly mean I regard them as synonymous. And if two people disagree regarding the confirmation conditions of a statement, that would imply they don't mean the same (or completely the same) thing when they express that statement, even if they use the same words.

It also makes sense for internal affairs. I make a statement about some internal affair, like "I see red", formally $See(i,r)\wedge Red\left(r\right)$. Here $i$ refers to myself and $r$ to my current experience of red. Then this is true iff there is some piece of evidence that $E$ which is equivalent to that internal statement, namely the experience that I see red. Then $P\left(See\right(i,r)\wedge Red(r)\mid E)=1$ if $E=See(i,r)\wedge Red\left(r\right)$, otherwise $P\left(See\right(i,r)\wedge Red(r)\mid E)=0$.

Again, the "I" here is logically an individual constant $i$ internal to the agent, likewise the experience $r$. That is, only my own experience verifies that statement. If there is another agent, who also sees red, those experiences are numerically different. There are two different constants $i$ which refer to numerically different agents, and two constants $r$ which refer to two different experiences.

That is even the case if the two agents are perfectly correlated, qualitatively identical doppelgangers with qualitatively identical experiences (on, say, some duplicate versions of Earth, far away from each other). If one agent stubs its toe, the other agent also stubs its toe, but the first agent only feels the pain caused by the first agent's toe, while the second only feels the pain caused by the second agent's toe, and neither feels the experience of the other. Their experiences are only qualitatively but not numerically identical. We are talking about two experiences here, as one could have occurred without the other. They are only contingently correlated.

Now then, what about the mixed case "I will see red"? We need an analysis here such that the confirming evidence is different for statements expressed by two different agents who both say "I will see red". My statement $\exists x\left(WillBecome\right(i,x)\wedge \exists y(Red\left(y\right)\wedge Sees(x,y)\left)\right)$ would be (now) confirmed, to some degree, by any evidence (experiences) suggesting that a) I will become some future person x such that b) that future person will see red. That is different from the internal "I see red" experience that this future person would have themselves.

An example. I may see a notice indicating that a red umbrella I ordered will arrive later today, which would confirm that I will see red. Seeing this notice would constitute such a confirming experience. Again, my perfect doppelganger on a perfect twin Earth would also see such a notice, but our experiences would not be numerically identical. Just like my doppelganger wouldn't feel my pain when we both, synchronously, stub our toes. My experience of seeing the umbrella notice is caused (explained) by the umbrella notice here on Earth, not by the far away umbrella notice on twin Earth. When I say "this notice" I refer to the hypothetical object which causes my experience of a notice. So every instance of the indexical "this" involves reference to myself and to an experience I have. Both are internal, and thus numerically different even for agents with qualitatively identical experiences. So if we both say "This notice says I will see a red umbrella later today", we would express different statements. Their meaning would be different.

In summary, I think this is a good alternative to the 0P/1P theory. It provides a unified account of meanings, and it correctly deals with distinct agents using indexicals while having qualitatively identical experiences. Because it has a unified account of meaning, it has no in-principle problem with "mixed" (internal/external) statements.

It does omit possible worlds. So one objection would be that it would assign the same meaning to two hypotheses which make distinct but (in principle) unverifiable predictions. Like, perhaps, two different interpretations of quantum mechanics. I would say that a) these theories may differ in other aspects which are subject to some possible degree of (dis)confirmation and b) if even such indirect empirical comparisons are excluded a priori, regarding them as synonymous doesn't sound so bad, I would argue.

The problem with using possible worlds to determine meanings is that you can always claim that the meaning of "The mome raths outgrabe" is the set of possible worlds where the mome raths outgrabe. Since possible worlds (unlike anticipated degrees of confirmation by different possible experiences) are objects external to an agent, there is no possibility of a decision procedure which determines that an expression is meaningless. Nor can there, with the possible worlds theory, be a decision procedure which determines that two expressions have the same or different meanings. It only says the meaning of "Bob is a bachelor" is determined by the possible worlds where Bob is a bachelor, and that the meaning of "Bob is an unmarried man" is determined by the worlds where Bob is an unmarried man, but it doesn't say anything which would allow an agent to compare those meanings.

It's worth noting that no reference to preferences has yet been made. That's interesting because it suggests that there are both 0P-preferences and 1P-preferences. That intuitively makes sense, since I do care about both the actual state of the world, and what kind of experiences I'm having.

Believing in 0P-preferences seems to be a map-territory confusion, an instance of the Tyranny of the Intentional Object. The robot can't observe the grid in a way that isn't mediated by its sensors. There's no way for 0P-statements to enter into the robot's decision loop, and accordingly act as something the robot can have preferences over, except by routing through 1P-statements. Instead of directly having a 0P-preference for "a square of the grid is red," the robot would have to have a 1P-preference for "I believe that a square of the grid is red."