You're basically right. Most of the bulbs you see on small OPE (chainsaws, hedge trimmers, blowers, whackers, etc) are indeed purge pumps (and not primers, despite what people call them). They push or pull fuel through the lines and carb, purging the air out. Makes for quicker starts. The downside of 'em is that they tend to develope air leaks after a while. I'm not a fan of them. Just my opinion, based on my experiences...
True "primers" on small OPE are usualy piston pumps that spray a shot of fuel into the carb throat or intake manifold as a means of enriching things for cold starting in place of a choke. You won't see that on much 'modern' small OPE. Back in 'the old days', McCulloch chainsaws were where you'd find the most primer systems. Many lawmowers do still have bulb type primers in place of chokes.
A purge pump can be pressed 100 times without flooding the engine. Once the air is purged, all you're doing is passing fuel through the lines and carb and back to the tank. The bulb feels 'harder' to push once the air's out.
A primer bulb or pump will flood the engine if it's pressed too much. For the most part, you won't find a primer on a piece of OPE with a choke. You will find a choke on a purge pump/bulb equipped piece of OPE, as it still needs some method of cold start enrichment.