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 

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.

This seems to be a sort of thing that requires a decent amount of trust. If anyone wants to try discussing a butterfly idea (whether emotions or novel economic models) in a safe space, I have done NVC (Non-Violent Communication) before for maybe 100 hours, talked to people about sensitive things and kept their secrets. (In case you look NVC up, the method is not made by rationalists and has some flawed assumptions; I don't intend to follow it anywhere closely). Feel free to send me a message.

Sure, I would be interested in participating in that (or possibly even organizing that).

How I imagine an online butterfly/intellectual-nurturance group working is as being a place where people can express their thoughts/feelings/ideas in the group setting, if they are comfortable with it, or meet others who would be interested in having a 1:1 conversation about their thoughts/ideas if they prefer that.

As someone who understands this idea all too well, I approve of this post.

I think my neuroatypicality might make me really good at "not crushing butterflies", but not so good at regular interactions with people, especially in groups or on the Internet. 
There are groups with different communication norms, I wonder if it would make sense to have an online "butterfly" group for rationalist-adjacent folk? Or is this is a kind of thing that's best handled through one-on-one interactions?

Shared frames are investments that allows us to coordinate better. This, I think, is a good frame. 

This allows me to express some of the thoughts I've had in a way that's clearer to myself and easier for other to comprehend, if they are familiar with the frame. For example: "Why would anyone pay for medical advice that isn't epistemically legible?"

An aside, I also found your post on butterfly ideas really insightful, though I'm biased by being a HSP with a significant portion of my mental models more-or-less being "butterfly ideas".

This feels insightful.

I have experienced incremental_progress-towards-a-clearer-model_of_reality, which feels like one of the highest praise one aspiring rationalist can give another.