I think your perplexity arises since the definition your cited is easy to misunderstand if you don't look at it carefully.
Let's try and break it in two parts:
A black body is a body which absorbs all the radiation it receives (or, equivalently, reflects none of it).
A black body is a body that gets heated by the energy it absorbs, and emits thermal radiation accordingly (indeed, black body radiation if it's in thermal equilibrium with the environment).
If you look carefully, it doesn't say that such a body is heated just by the radiation it absorbs. It says energy.
In the specific case, a powered-on light bulb get just a very tiny fraction of the energy that heats it up in the form of radiation from the external evironment. All the rest (almost all) is supplied by the means of an electric current carried by a wire: the electrons strike the tungsten atoms making them to vibrate furiously, and you know heat is just kinetic energy.
This explains why the filament (NOT the whole bulb) is a good approximation of a black body. It's not perfect though, as other users mentioned, since it's not a perfect absorber (perfect absorbers do not exist in nature, as far as we know).
Note that the light bulb filament is no longer a good approximation of a black body once you turn it off.
Another (even better) good approximation of a black body is the surface of the Sun (or of a star whatsoever). It does not certainly emits so much EM energy because it's heated by the radiation absorbed from other sources (that is present nonetheless). Rather it is heated by the fusion occurring hundreds of thousands kilometers below.