In security engineering, a trusted component of a system is a component that has the ability to violate the system's security guarantees.  For instance, if a security engineer says "Alice is trusted to guard the cookie jar", that means "Alice has the ability to render the cookie jar unguarded".

I notice that the four examples at the beginning of this post all seem to slot pretty nicely into this definition:

• "I decided to trust her" => I assigned Alice to guard the cookie jar by herself
• "Should I trust him?" => Should I allow Bob to guard the cookie jar?
• "Trust me" => Please allow me to access the cookie jar
• "They offered me their trust" => They granted me access to the cookie jar

If you think about them as being about security policies, rather than epistemic states, then they seem to make a lot more sense.

I think the layperson's informal concept of "trust" is more muddled than this, and conflates "I'm giving you the power to violate my security" with "I am comfortable with you having the power to violate my security" and maybe some other stuff.

This feels kinda straw-vulcany, sort of missing the point about what people are often using trust for.

I'm not actually sure what trust is, but when I imagine people saying the sentences at the beginning, at least 35% and maybe 75% of what's going on is more about managing a relational stance, i.e something like "do you respect me?". 

I do expect you'll followup with "yeah, I am also just not down to respect people the particular way they want to be respected." 

So a major part of how I handle this sort of thing is usually conveying somehow "I don't believe that, and generally don't trust your epistemics enough to believe-you-by-default in this domain, but, also, I [respect, in whatever way I do happen to respect] the person." (Usually this is managed more by vibes than by words, though if I found myself in a situation where someone I cared about said "trust me" it probably means something is fragile enough to require some careful linguistic work as well)

The main exception here is self-fulfilling prophecy, but that’s not obviously centrally involved in whatever “I decided to trust her” means.

I think it might be.

I think the core of the contractual interpretation is right, but I'd phrase it differently. I think "deciding to trust" someone implies a deal where you ally yourself with them in a way that leaves you vulnerable to hostile actions from their side – you're choosing "cooperate" in the Prisoner's Dilemma in expectation of greater mutual gain. In exchange, they (1) commit not to take said hostile actions, (2) potentially "reciprocate trust" by showing symmetrical vulnerability (either to ensure MAD, or because pooling resources improves your ability to work together), and (3) bias their policy towards satisfying your preferences, which includes being more responsible when interacting with you and giving you some veto on their actions (and if they fail that, they've "betrayed" your trust).

And there's some degree of acausal culture-based negotiation going on. E. g., it's not that people explicitly track all of the above consciously, it's that we have subconscious pre-computed scripts of "if someone utters 'I trust you' and your model marks them as an ally, modify the prior used by your plan-making processes towards satisfying their preferences more". Which is implemented by automatically-activated emotional responses; e. g., the feeling of responsibility automatically flaring up when you're thinking about someone who put their trust in you.

So to some extent it is a self-fulfilling prophecy. If you've decided to trust someone, and their learned-instinct systems recognize that, they'll automatically reciprocate and become more trustworthy. (Or so the underlying calculations go. It might not work: if they jailbroke out of these systems, if they've never had them installed (psychopaths), or if other systems overrode that.)

Not that there isn't all kinds of sloppy language involved, e. g. when "I trust her" actually just means "I've evaluated that the process she uses for [whatever] is reliable" or "leaving myself vulnerable here is a risk, but the CBA is in favour", instead of "I've initiated the 'trust' social dance with her and I don't think she's defecting".

My favourite definition of trust is “willingness to be vulnerable” and I think this answers most of the questions in the post. For example it explains why trust is a decision that can exist independently from your beliefs: if you think someone is genuinely on your side with probability 95%, you can choose to trust them, by doing something that benefit you in 95% of cases and hurt you on the 5% of cases, or you can decide not to, by taking actions that are better in the 5% of cases. Similar for trusting a statement about the world.

I think this definition comes from psychology, but I also found it useful when talking about trusted third parties in cryptography. Also in this case, we don’t care about the probability that the third part is malicious, what matters is that you are vulnerable if and only if they are malicious.

To me, the phrase “I decided to trust her” throws an error. It’s the “decided” part that’s the problem: beliefs are not supposed to involve any “deciding”. There’s priors, there’s evidence, and if it feels like there’s a degree of freedom in what to do with those, then something has probably gone wrong.

Trust isn't about having a particular belief, it's about acting on a particular type of hypothesis despite lack of convincing evidence supporting it.

"I decided to trust her" does not mean that you believe that she absolutely won't do some bad X, but that you decided to act as if she won't. That is, you were prepared to take the risk.

Likewise to "trust me" doesn't mean believing that I will never do a bad thing, but to take the risk despite the lack of sufficient evidence.

1.  "Trust" does seem to me to often be an epistemically broken thing that rides on human-peculiar social dynamics and often shakes out to gut-understandings of honor and respect and loyalty etc.

2. I think there is a version that doesn't route through that stuff. Trust in the "trust me" sense is a bid for present-but-not-necessarily-permanent suspension of disbelief, where the stakes are social credit. I.e. When I say, "trust me on this," I'm really saying something like, "All of that anxious analysis you might be about to do to determine if X is true? Don't do it. I claim that using my best-effort model of your values, the thing you should assume/do to fulfill them in this case is X. To the extent that you agree that I know you well and want to help you and tend to do well for myself in similar situations, defer to me on this. I predict you'll thank me for it (because, e.g., confirming it yourself before acting is costly), and if not...well I'm willing to stake some amount of the social credit I have with you on it." [Edit: By social credit here I meant something like: The credence you give to it being a good idea to engage with me like this.]

Similarly:

• "I decided to trust her" -> "I decided to defer to her claims on this thing without looking into it much myself (because it would be costly to do otherwise and I believe-- for some reason-- that she is sufficiently likely to come to true conclusions on this, is probably trying to help me, knows me fairly well etc.) And if this turns out badly, I'll (hopefully) stop deciding to do this." 
• "Should I trust him?" -> "Does the cost/benefit analysis gestured at above come out net positive in expectation if I defer to him on this?"
• "They offered me their trust" -> "They believe that deferring to me is their current best move and if I screw this up enough, they will (hopefully) stop thinking that."

So, I feel like I've landed fairly close to where you did but there is a difference in emphasis or maybe specificity. There's more there than asking “what do they believe, and what caused them to believe it?” Like, that probably covers it but more specifically the question I can imagine people asking when wondering whether or not to "trust" someone is instead, "do I believe that deferring these decisions/assumptions to them in this case will turn out better for me than otherwise?" Where the answer can be "yes" because of things like cost-of-information or time constraints etc. If you map "what do they believe" to "what do they believe that I should assume/do" and "what caused them to believe it" to "how much do they want to help me, how well do they know me, how effective are they in this domain, ..." then we're on the same page.

I've always felt basically the same about trust. It's nice to see that I'm not the only one.

I'd add/emphasize that trust is really a 2-place word. It doesn't really make sense to say "I trust Alice". Instead, it'd make sense to say "I trust Alice to do X" or "I trust Alice's belief about X" or "I expect Alice's prediction about X to be true". Ie. instead of trust(person), it's trust(person, thing).

(And of course, as mentioned in the post, it isn't binary either. The output of the function is a probability. A number between zero and one. Not a boolean.)