That’s a very broad question. Just to offer some brainstorming of assorted random possible reactions to it, since there’s no one answer:
“Villain” is a word, which itself has various meanings to different people in different contexts.
No, I’d say that villains are not “basically just bullies”. My first impulse response to the question would be that:
* I think of (and have experienced) bullies as people who are usually internally fearful and insecure, and have developed an (often subconscious, developed in childhood) habit of compensating by projecting a false ego identity of a formidable menacing person looking for conflict (but actually trying to avoid it by preemptively seeming intimidating and behaving aggressively… towards people they think/hope won’t resist). Bullies tend to be relatively easy to reduce to cowards by actual strong resistance. They tend to operate in a strategy of “bully or be bullied, or avoid or submit to people you can’t bully”, and assume most other people do too.
* I think of villains as either A) dramatic caricatures invented by authors to provide an entertaining fantastic arch antagonist, such as a Bond villain, evil magic user, or other exaggeratedly dastardly evil scoundrel, or B) someone who does really awful and destructive things.
Many literary and cinematic villains lack realistic psychology and history that realistically explains how and why they do what they do.
As for different types, well there are tons of academic and pop journalism articles with theories organizing fictional in various ways, as if those are the only types, which to me seems to unfortunately lead to more people creating more cookie-cutter derivative villain characters who then tend not to be very interesting because they come from a stale confining mindset of “we need a villain characters so we’ll pick a type we know and fill in the blanks and maybe add a twist”, instead of developing an actual human-like character.
There are also plenty of people who try to categorize real-world villains, whether by what laws they break, what they do, their social position, or whatever. Again, I’m not sure how particularly valuable that is. I think it can get you only so far, but really they are humans, and the best understanding of each one is probably through relating to them as humans.
Real-world villains, like most humans, usually have some trauma, but the type or severity or other circumstances lead to those people doing horrible enough things to be considered villains by others. Often, like bullies, it’s ultimately a projection of that trauma, with an ego created to avoid facing whatever they dread most from that trauma. That gets many people into a self-reinforcing loop behavior, where they in some form do the thing that traumatized them, and the part of them that loathes themselves for that, reinforces the behavior that covers it up. That can and often does develop into escalation of the hostile behavior.
