The word An means, among other things “to place the hand (or hands) on,” or “to restrain, to control.” In daily speech, it is used as a verb for pressing a button, and in various compounds for taking someone’s pulse (anmai), or for massage (anmo), etc. The conventional translation “push” is not really adequate, in my opinion, because the application of An does not necessarily imply a push. As for the form posture An, as part of the Grasp Sparrow’s Tail sequence—while the ending posture may resemble a push, the form itself includes the drawing back of the body and the arms prior to extending them forward. So An encompasses more than a push. I’m inclined to think that the “control” connotation is more in line with the meaning of An as a root configuration of taiji jin than “push.” It's making contact with the hands in order to maintain control.
Regarding the term Push Hands, yes, that is a reasonable translation of tuishou. There is reason to think, however, that tuishou is a relatively modern term for the exercise. Just speculating, it could be a sort of nickname based on what it looks like. Earlier terms were da1 shou (join hands, match hands), and da3 shou. Da3 is kind of a multi-purpose verb, but da3 shou3 most often means something like sparring.