I’ll pushback a bit here. Every mixture model can be interpreted as a single, fixed but unknown group index common to all observations or group indices that very from observation to observation, which means every one of these models has the same ambiguity. In many fields one of these options often becomes convention, but that doesn’t mean that it becomes a general truth!
Again when constructing posterior inferences the difference between the two perspectives doesn’t matter, but when you want to make retrodictions or predictions the two perspectives diverge because retrodictions and predictions depend on more details of the assumed generate process. I think that it’s much more clear to explicitly specify which of those processes you’re assuming instead of relying on an assumed notational convention that may or may not be understood by others.