Sorry if this comes off a bit pedantic but ...
I believe your asking of the question answered
your own question
Not pedantic, old bean, perceptive. (I have a pretty good handle on what I think and do regarding these transactions but like asking questions to test the waters, so to speak, and to see what I can learn from the folks talking around the campfire.)
That said, I am less sanguine than some about the restraint of various government agencies regarding private property, search and seizure, datamining, and grandfathering various types of firearms and ammunition. So I tend to err on the side of simple bill of sales rather than more official records.
And like others have commented regarding taxation/stamp tax, don't get me started....