soft question - Theorem versus Proposition

Of course, this is a very subjective question, but I would tend to use "Theorem" only for a statement which has genuine content (whether my own, or one I am citing) and which I wouldn't expect the reader to be able to prove themselves fairly easily. Usually a paper shouldn't have many of these, probably no more than one per section.

"Proposition" I would use after having given a definition, when showing that some fairly straightforward (but not completely obvious) consequence holds; for instance showing that some linear subspace of functions is actually a subalgebra. This is probably close to how you said you use "claim", although I suppose the difference is that you can propose something somewhat out of the blue following a definition, while "claim" is usually directly related to some logical structure which is already moving forward, say to highlight a point midway through the proof of a theorem.

So I make the distinction that Proposition is something that the reader, if so inclined, could easily prove for themselves once they understand the definition. It highlights a result that could just as well have been stated in plain text, emphasizing that while it may be straightforward to prove, it is nevertheless worthy of note.