As I understand it, Protagonist is the one trying to achieve something. The Antagonist is the one trying to hinder him. This makes a lot of "bad guys" be Protagonists. In Transformers, the Decepticons would often be the Protagonists. This makes James Bond an Antagonist.
It's all in how you position the goal. If you say the goal is to Stop the Decepticons from doing "fill in evil deed here", then the protagonists are the ones pursuing that goal, and the Decepticons ones trying to prevent or avoid that.
Keep in mind that the Story Outcome is tied to the Story Goal. This is a good indicator as to how the author wants the audience to understand who the protagonist and antagonist are.
Why do villains tend to be the impact characters in horror stories? Also, are there any examples of horror that come to mind in which the Influence Character is not the Villain or Antagonist?
The Villains, who are also often the Antagonists, are frequently the Influence Character because the IC is one of the few characters whose perspective must exist until the end of the story. So the Villain is often the IC in horror stories because everybody else is dead or removed from the scene.
There are a couple of movies that come to mind where the Villains (or monsters) are not the IC, though several of them are mixed genres such as Horror/Fantasy, Horror/SciFi, or Horror/Drama"
The Reaping: IC: The little girl; Villain: Doug (and the townsfolk)
Pitch Black: IC: Riddick; Villians: Monsters
Aliens: IC: Newt; Villains: Aliens/Mother Alien
Stir of Echoes: IC: Ghost (Samantha Kozac); Villians: The neighbors that killed her and covered it up.
Edge of Darkness: IC: Hit Man; Villains: Corporate and government cover-up guys
30 Days of Night: IC: Stlla Oleson (love interest); Villains: Vampires
Granted, these are the exceptions rather than the rules, and each of these seem to be more than just "horror" movies.
How do they relate to the Protagonist and Antagonist?
In the strictest sense, the protagonist is for the goal and the antagonist is against the goal, so who is who is completely dependent on how the author defines what the goal is. If the goal of Rosemary's Baby, for example, is to bring the Devil's child successfully to this world, then the cultists would certainly be for that. If the goal of The Stepford Wives was to make all wives complacent, agreeable, and controllable, then the men of Stepford would be in the protagonist camp.
HOWEVER, part of what is NOT explicit is the commentary or "spin" that the author puts on the story. Is the author for the son of the Devil, or against it? Is the author for marital automatons, or against them? The author's intent is usually built into the goal. It is obvious in some cases, and not so obvious in others.
In The Stepford Wives, the author seems to lean in favor of preventing or avoiding the outcome -- thus The Stepford Wives is a Failure story because Joanna (played by Katherine Ross) becomes a Stepford wife, and therefore the story acts as a cautionary tale.
In Rosemary's Baby it's not so obvious and leaves room for interpretation. I believe Mark Harrison (the Dramatica story analyst) chose the pro-devil way of looking at it (Story Outcome = Success) because it seemed more consistent with the time in which the story was written/filmed, as well as the sensibility of the author/director (Ira Levin/Roman Polanski). This is not to say that the author(s) was for bringing the devil's child to our world, but more likely that he put that message out there to shock audiences' sensibilities and create moral outrage.
There's a fine line between being straightforward by saying what you mean and saying something inflammatory to provoke discussion, but the important aspect of it from an author's point of view is to be consistent. Pick what the goal is and what you want to say about that goal and then place the overall story characters appropriately in relationship to it.
The differences between the Contagonist and the Antagonist are best understood by looking at their component character elements.
At the motivation level, the Contagonist represents Temptation and Hinder, whereas the Antagonist represents Reconsider and Avoid/Prevent.
Where a contagonist might lead one astray and impede one's progress, an antagonist would make the argument that one should completely reconsider one's plans while actively acting to avoid or prevent them from taking place.
The same type of relationships between the Contagonist and Antagonist exist in the other three levels of character elements -- Methodology, Purposes, and standards of Evaluation.
Don't they mean the same thing?
Do not confuse an archetypal Antagonist in Dramatica with a classical "Villain." A Villain is a non-Dramatica term often used to describe a character responsible for causing or doing things the audience doesn't like (i.e. the 'Bad Guy') -- usually to characters the audience likes. A Dramatica Antagonist, on the other hand, is the character trying to prevent the overall goal from coming to pass or being achieved. Quite often characters are both the Antagonist and the Villain (e.g. Dr. Evil in Austin Powers, or the Terminator in The Terminator). Other times an Antagonist will be more complex than a simple Villain (Jane Hudson in What Ever Happened to Baby Jane?, or Buddy in Lonestar), or a Villain may be the Protagonist (Richard III in Shakespeare's Richard III).