The map is not the territory.
Our understanding perception of reality is not fully accurate. This is important to understand. Not being aware of it, assuming that you are percieving reality as it is when in fact you aren't, is a source of error.
But our understanding of our model of reality is not fully accurate either. Not being aware of that is also a source of error.

Reality is the territory.
Our mental model of reality is the map.
What we think our mental model of reality is, is the map of the map.

We have mental models of other people's minds. We also have a mental model of our own mind. It is not fully accurate.

Specifically, I want to talk about concepts.
A word/concept has two different meanings, what we think it means to us (map of map), and what it actually means to us when processed by our brain (map). (while the similarity cluster of reality that the concept is meant to capture would be the territory).

Or, a different way to put it, a concept has two different interpretations depending on which mental machinery we use to process it, and yet we aren't automatically aware of that.

As you can imagine, this can cause all kinds of errors.

Tabooing helps. What do you do when a word doesn't mean to your brain what you think it means, or when you are aware its meaning is somehow ambigous/contradictory/misleading? Remove the word from the equation entirely, and replace it with actual meaning you want to use.
Simply saying, "I define 'human' to mean 'featherless biped'" is not good enough - actually remove the word entirely, in order not to contaminate what you are saying with whatever meaning our brains already have attached to the tabooed word.

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[-]TLW20

Question: is this truly at the language level? Or is this beneath it?

Could someone who is fluently bilingual (to the point of being able to think in either language) get a more accurate view by repeating the same thought chain in both languages and noting differences in their own reaction?

I am fluent in more than one language and I think in different languages sometimes, usually when one of the languages has words than better map onto the concepts/ideas I'm trying to think about.

[-]TLW10

Is this more accurate? Or merely faster?

It definitely feels differently. The words feel different. Not all words translate precisely to a different language, some words are unique to languages, and many words have different "flavors" in different languages that don't exactly translate.

So my mental model of it is, when I think in different words, my thoughts do mean something different.

I suppose I can't know for sure if it's always more accurate. Sometimes I'm just in a mood to think in a particular language, maybe to vary my perspectives a bit. (I am fluently trilingual)

Sometimes I do switch languages mid-thought because I just feel a certain word/idea needs to be said in particular language, in which case I am reasonably certain it is more accurate.

[-]TLW10

Sometimes I do switch languages mid-thought because I just feel a certain word/idea needs to be said in particular language, in which case I am reasonably certain it is more accurate.

How would you distinguish between 'it is more accurate to think this thought in language X than language Y' and 'language X is a more accurate translation of the underlying thought than language Y'?

How would you distinguish between them? 
They seem "the same" to me, pretty much

'it is more accurate to think this thought in language X than language Y'
seems to imply that the thought is created as I think

'language X is a more accurate translation of the underlying thought than language Y'
seems to imply that there is already a "wordless" thought that's waiting to be expressed

Now that I try to "feel" a difference 

[-]TLW10

'it is more accurate to think this thought in language X than language Y'
seems to imply that the thought is created as I think

'language X is a more accurate translation of the underlying thought than language Y'
seems to imply that there is already a "wordless" thought that's waiting to be expressed

That's pretty close to what I'm wondering. Though I would perhaps phrase it more as:

'it is more accurate to think this thought in language X than language Y'
'thinking a thought in language X' means 'forming the thought in language X'

'language X is a more accurate translation of the underlying thought than language Y'
'thinking a thought in language X' means 'forming the underlying "wordless" thought and translating it to language X'

 

How would you distinguish between them? 

I don't know if there's any simple answer here. It might be possible to distinguish them statistically. 

Consider the case of "thinking in language X, speaking it in language X, then speaking it in language Y". Several cases:

  1. If 'thinking a thought in language X' means 'forming the thought in language X':
    1. You think the thought in language X.
    2. You speak the thought in language X.
    3. You translate the thought directly from language X to language Y.
    4. You speak the thought in language Y.
  2. If 'thinking a thought in language X' means 'forming the underlying "wordless" thought and translating it to language X', possibility A:
    1. You think an underlying wordless thought.
    2. You translate the wordless thought to language X.
    3. You speak the thought in language X.
    4. You translate the wordless thought to language Y.
    5. You speak the thought in language Y.
  3. If 'thinking a thought in language X' means 'forming the underlying "wordless" thought and translating it to language X', possibility B:
    1. You think an underlying wordless thought.
    2. You translate the wordless thought to language X.
    3. You speak the thought in language X.
    4. You translate the language X thought directly to language Y.
    5. You speak the thought in language Y.

These three possibilities have different sets of thought translations. If you have a measure of 'accuracy of translation', it may be possible to distinguish these[1].

  1. ^

    Especially if someone can think in >2 languages. It may not be possible or practical with 2.