But generally I agree, animations are easily overused. They're quite useful for showing (and sometimes hiding) UI elements, as this leads to a interface devoid of shock and surprise. Heck, I think I read something about that in a paper that predates the web…
Last bit is interesting, animation can convey useful information, but mostly they're attention grabber. They have a place, but care is required.
[0] Source: https://en.wikipedia.org/wiki/Sic
As a sidenote.. I'm currently working on a patch to CyanogenMod 12.1 that allows you turn off fancy animations (such option exists already when you turn on power save mode)
Where should animation libraries be used? I'd think they should be used where a user would appreciate an animation. Such places could be while waiting for something to load, or maybe on a website dedicated to animation libraries. I think that would be an appropriate place for animations. It'd be weird if a website dedicated to animations didn't have animations on it.