There's a trend for the latter category to be renamed Free Kriegsspiel Revolution (see https://boardgamegeek.com/thread/2500148/ancient-roleplaying... for a writeup). Kriegsspiel is a Prussian term for refereed war games, developed to enable better training for their army; the original free kriegsspiel movement arose because the rules felt too constrained.

These are not really all satires, although I like that post about satirical games a lot.

The Quick Primer is about as good a general definition of OSR as you're going to get.