Compil[e/er/ation] Error
Looking at the number of results on Google Books and Google Scholar, and also Google:
+--------------------+------------+---------------+---------+
| |Google Books|Google Scholar |Google |
| | | | |
+--------------------+------------+---------------+---------+
|compiler error |7,480 |1,950 |386,000 |
+--------------------+------------+---------------+---------+
|compile-time error |6,620 |1,850 |311,000 |
+--------------------+------------+---------------+---------+
|compilation error |5,260 |1,620 |1,660,000|
+--------------------+------------+---------------+---------+
|compile error |2,280 |710 |2,550,000|
+--------------------+------------+---------------+---------+
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:
compiler error: an error detected by the compiler, an error message produced by the compiler, an error within the purview of the compiler. Not an error in the compiler.
compilation error: not an error in the compilation (the compilation did what it was supposed to do), but an error found during compilation, etc. Same as above.
compile-time error: an error found at compile-time (cf. "runtime error")
compile error: …
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.