Bottom line: It's just down to personal opinion, at the end of the day.
I always thought "dark" in this sense this meant stylistic and character changes, perhaps tending more towards the horror genre.
This is essentially correct. Not necessarily trending towards horror, but more serious and more troublesome for sure. The events which transpire are grittier, more emotional, and and more "badass." Some might also say more realistic, although that depends heavily on the property in question.
For example, the Nolan trilogy of Batman movies are generally considered "darker" than the previous Batman series (at least darker than Batman Forever and Batman & Robin, anyway. The Tim Burton films were pretty dark in their own way, although it's definitely safe to say Heath Ledger's Joker was orders of magnitude darker than Jack Nicholson's Joker, for example.)
The Nolan films are still action movies at the end of the day, though, nothing especially "horror genre" about them.
Or perhaps the producers opening the door to allow bad things to happen to actors.
Well, hopefully nothing bad happens to the actors. ;)
It's a safe bet that whatever happens to the characters will be more intense, though.
Another example is that The Empire Strikes Back is often considered the "darkest" of the original Star Wars trilogy. Presumably because the good guys lose.
Definitely. Han Solo being encased in carbonite, Luke losing his hand. Some very dark things happened in ESB. Of course, as a whole it's hardly the "darkest" movie ever made, but in comparison to ANH and ROTJ, for sure it's the darkest.