What is the name of our Solar System? what is our Solar System called rather than "the Solar System"?

I've found "Sol System" and "Monmatia", too, but is there more to it?
It's just called "the Solar System". (Plenty of places and objects have names like that; it's no different from "the Arctic" or "the Moon" or "the Sun".)

("Sol system" is an invention of science fiction writers; it has no general use outside some science fiction contexts. Anything else is going to be something similar, or general crackpottery of one kind or another.)