That seems customary, yes.
Note that the body diode is already drawn in the symbol, that's the arrow pointing into/out of the center. (At least, it is for old school lateral MOS, where the substrate and source and distinct elements. Using it for modern VDMOS is kind of weird.) So the antiparallel diode is redundant.
And yeah, there is a continuum from zener (tunneling) action to avalanche (impact ionization) breakdown, the terms are used fairly interchangeably as a result.
Note also that, although a transistor might be rated for avalanche, that doesn't mean it's a good idea to use it that way. It takes time for the avalanche current to dissipate, greatly increasing device dissipation when used at high supply voltages. Abusing avalanche ratings in a switching converter, for example (where stray inductance causes flyback spikes), will likely lead to quick failure, even if the device is apparently well within ratings.
There's one more "gotcha" about avalanche ratings: it is exclusive of good MOSFET performance. If you need that one tiny bit of extra performance (particularly in body diode recovery time), you'll have to keep shopping. (Probably not something a beginner will worry about, at least!)
Tim