I can’t remember who said this first, and I can’t remember if I’ve already put this on the blog, but the following definition may be helpful:
Every statistician uses Bayesian inference when it is appropriate (that is, when there is a clear probability model for the sampling of parameters). A Bayesian statistician is someone who will use Bayesian inference for all problems, even when it is inappropriate.
I am a Bayesian statistician myself (for the usual reason that, even when inappropriate, Bayesian methods seem to work well).
(The above is perhaps inspired by the saying that any fool can convict a guilty man; what distinguishes a great prosecutor is the ability to convict an innocent man.)