If I didn't know you better, I'd worry you were being a touch willfully obtuse here. (I say this playfully!)
Ok, so to the basics:
- Democrats combine to form the Democratic Party, or the "Dems"
- Republicans combine to form the Republican Party, or the "GOP" (which is about whimsy folksy as one gets these days, but I digress)
I wouldn't say "Republic Party" any more than "Democrat Party". It's not the name, right?
Yet some of the pettier elements in MAGA have seen fit to use the name "Democrat Party" because in their minds it's a way of insulting them by suggesting they are not "democratic" or something. It's juvenile and stupid.
I think it's perfectly fine to name your party whatever you want. I sort of roll my eyes at the "Freedom Caucus" but I'm not going to make nicknames for them. I don't even indulge in "Gym Jordan", as much as he deserves it.