This is my understanding, but it appears we're not correct. A cave dweller is a troglodyte, but a troll is not specifically a monster under a bridge. In fact, an old English term for walking is to troll [0]. It would appear that this is where the terms stroll and trolley come from.
A troll is not specifically the monster under the bridge, no. The troll under the bridge is from one fairy tale about trolls, and I find that the closest use of troll to what Internet Trolls do.
But no, trolls in general do not live under bridges. Agreed.
[0] https://en.oxforddictionaries.com/definition/troll
[1] https://www.etymonline.com/word/troll