jk rowling’s reasoning as to why fenrir greyback turned remus into a werewolf: remus’s father insulted him so he did it as an act of revenge
the actual reason greyback bit remus: the temptation to succumb to the fact that biting remus whose name literally means ‘werewolf’ would be the greatest feat in lycanthropic irony the world had ever seen
Remus means swift. So no, it does not literally mean “werewolf”. Also if this is an allusion to the founding myth of Rome it is still wrong.
“Rhea Silvia conceives the twins by the god Mars, or by the demi-god Hercules; once the twins are born, Amulius has them abandoned to die in the river Tiber. They are saved by a series of miraculous interventions: the river carries them to safety, a she-wolf (in Latin, lupa) finds and suckles them, and a woodpecker feeds them. A shepherd and his wife find them and foster them to manhood, as simple shepherds.”
So Remus isn’t a werewolf.