Ellipsis is the technically correct word to use when talking about the omission of words. I don’t think it’s going to hurt anything to use a less technical term like “omission” or “implied word”.
I do think using “elide” or “elision” when you really mean “ellipsis” would be harmful, because in a linguistic context “elision” has a distinct meaning.
Being sloppy about the usage of technical terms can make things difficult for learners already saddled with terms that are sometimes unique to EFL courses. It doesn’t help that the words seem similar and are both about omissions.
In linguistics, an elision or deletion is the omission of one or more sounds (such as a vowel, a consonant, or a whole syllable) in a word or phrase.
In linguistics, ellipsis (from the Greek: ἔλλειψις, élleipsis, "omission") or an elliptical construction is the omission from a clause of one or more words that are nevertheless understood in the context of the remaining elements.
The ellipsis on the main site has over 480 questions already, the elision tag has under 20 (and I found at least one of those tagged incorrectly). The description of those tags were added in 2013, so there is some history there worth considering.