Is there a specific arteritis called carotid arteritis?

Yes: You are probably asking about takayasu's arteritis. This is a large vessel vasculitis that can often involve the carotids.

Problem vs. Location: -itis at end of medical descriptive word means inflamation. Thus, arteritis: inflamation of artery, by any mechanism. Carotid: the larger arteries in our neck; supply blood to our brain & face. The inflammatory cells are our white blood cells. Usual dominant arterial disease is an accumulation of wbcs in the walls of arteries, typically starting childhood, thus inflamation. There are other forms.