What other names are used for a three-quarter shot?

First, those terms aren't exact synonyms. A 3/4 shot is any image where subject has been cropped at around the knees. American shot is sometimes used to specifically refer to composition where several subjects in interaction (e.g. partners in dialogue) have been cropped that way; and sometimes, it is indeed used more loosely as a synonym for 3/4 shot.

Richard W. Kroon's A/V A to Z: An Encyclopedic Dictionary of Media, Entertainment and Other Audiovisual Terms offers terms medium length shot and medium full shot as alternative to three-quarters shot.

The same dictionary suggests cowboy shot could be used when referring to an American shot. Both terms are inspired by the technique being frequently used American western movies in order to keep both face and gun (on belt) in the frame. Other sources, such as Ian Freer's list of camera shots or Videomaker, disagree and consider "cowboy shot" to mean depiction of a duel situation where one of the protagonists is very close to camera, large part of body cropped away but gun prominently displayed, and another is visible further away.

I'd probably call them knee-length portraits (example of the term used here).