There are noted differences between the two games. The best example I can come up with (thanks to the recent changes to the okami questions) is this question, which asks about a mini game that is not present in the PS3 remake.
A question asking about the absence of that minigame would not apply to the original release of the game, only the remaster. Thus, it would require the okami-hd tag. As mentioned in Robotnik's answer here and fleshed out here, it is perfectly acceptable to have tags with only one or two uses.
Thus, I don't think okami-hd should be turned into a synonym of okami. However, I think that the okami-hd tagged questions may need to be cleaned up a bit, since not all of them address remaster-specific features (as I've noticed you've been doing lately).