There's always going to be subjectiveness in your audience, even with the most black and white of works. But I would define a villain as a character who's actions and intentions are portrayed as negative or harmful in the context of the story, and are in some sort of direct opposition to another force or character(s). They're usually antagonists, and the term usually implies them to be a genuinely bad person.
It's important to note that there's not necessarilly a strict separation between villain and anti-villian. Anti-villains are called such because they lack personal qualities that we generally associate with villains, but are still filling the role of a villain. They may be doing the wrong thing for the right reason, but they're still doing the wrong thing.
And I think that highlights the fact that there's really two functioning definitions for what qualifies as a villain. There's the villain as a literary role, and villain as a personal description. An anti villain lacks villainous qualities by definition, just as an anti-hero lack heroic qualities by definition. But they're still functioning by that role based on their actions and their place in the story. But those definitions wouldn't work if we didn't have things that we consider to be innately heroic and villainous qualities. Heroes are brave, care for others, have a sense of justice, villians are selfish, lack compassion, are cruel - those are personal descriptions, but obviously don't apply to all characters who fill a heroic or villainous role.
There is an area, around the time that you slip into a morality that that doesn't include a "white", a "black", or either, that it becomes difficult to define these roles, or functionally irrelevant. Not everyone rivaling a hero is necessarily in a villain role, and not everyone rivaling a villain is necessarily in a hero role.
Protagonist and antagonist are really more about perspective. Protagonists are who your story is largely about. It's either from their perspective, or follows their actions or personal events. Antagonists are the people that oppose them. You can very much have protagonists that are genuinely villains, and antagonists that are genuinely heroes (I'm mostly meaning by literary role here). That being said, it's actually pretty rare, and I'd say most examples that people cite as villain protagonists are not clearly what most would call a villain. Most of those examples are from stories that displays morality lacking in one or both of the extremes (black/white), so they tend to be in a fuzzier area to start with.