I agree with pretty much all of your points, although I feel there's a bit of a nuance between performing a Reality Check as opposed to a check to see if you are dreaming. A reality check, depending on the perspective of the listener/user can both refer to the act of checking if you are not dreaming and the act of checking to see if you are. By contrast, checking if you are dreaming only refers to one act and is more specific.

Whether the ambiguity associated with Reality Check as a term is pro or a con, I don't know. Same goes for the specificity of checking if you are dreaming. What I do know is that, despite the fact checking if you are dreaming effectively functions also as checking if you are not dreaming, there are actually ramifications to your approach and with regards to your expectations based on which act you feel you are more chiefly concerned with performing. One is more to determine if you are indeed awake, the other if you are asleep and in a dream.

Truthfully I think checking to see if you are awake is likely to have more negative results because when we make habits of doing certain things, what we experience and determine ourselves to have experienced becomes associated with that habit. In other words, if you perform RCs in order to confirm you are awake or that you are not, in fact, dreaming, it stands to reason that, because you will inevitably perform more RCs while awake than while sleeping and dreaming, you will grow to expect the result of determining that you are actually awake. At that point, if performing the RCs aren't an active detriment to becoming lucid, at the very least, the fact you're pretty much just going through the motions for no real reason makes the practice pretty much pointless, no?

So, I guess in the end I agree more with calling it Dream Check or saying you are checking if you are in a dream better. Then again, I'm sure somebody whose capable of framing it right from the opposite perspective could make essentially the same argument I did, and it would be just as equally valid. Lol, depends all on how you view things I guess.