But when we say "an estimator is asymptotically normally distributed", what does it mean?
Using similar language to your first sentence, when we say an estimator is asymptotically normally distributed, we mean something like as the sample size increases, the sampling distribution of a suitably standardized version of the estimator converges in distribution to some particular normal distribution.
Are "central limit theorem" and "asymptotically normally distributed" synonymous?
Not in general, I think. Some quantity may be asymptotically normal but not come about as a result of any of the versions of the CLT (at least not in any obvious way - it might perhaps be that all of them can ultimately relate to the CLT, but I suspect it's possible to construct cases that would not).
However, very many estimators can be cast as a kind of average of some random variable and in that case a CLT-type argument may be indeed possible.
In some other cases you can combine the CLT with some other result to produce an argument that some estimator should be asymptotically normal (so the CLT may be involved but doesn't stand alone as the basis for the asymptotic normality).