Which one should I use of the following:
I think the "Compilation Error" is way too long for everyday use, even though it's the "correct" term. So, which one?
Looking at the number of results on Google Books and Google Scholar, and also Google:
Books and Scholars show a clear order of preference, for compiler error over compile-time error over compilation error over compile error, but unfortunately the general public seems to show the opposite tendency.
Grammatically, all four (or at least all but the last one, which is ironically the most popular) are fine:
Google n-gram viewer tells a similar story (click on the image), though, as if underscoring its unreliability, compile-time error seems to be missing from its corpus for some reason (zero results).
[Aside: broadening the date range gives a few results in 1890–1910… what's with that?]
I'd prefer compilation error myself, as it is the "correct" term, as you say, and only a few letters longer than the alternatives. Compiler error would mean that there is a problem with the compiler, rather than the source code. Compile error sounds a bit baby-talk to me.