No need to apologise for being a pedantic old scrote I'm one myself!
So I'm told.
Yes to the sprig flat side for better holding - panel pins have a head which means a very local point of contact and they take more force to tap in. That sprig hammer is a novelty! The point about sprigs is that they are easy to tap in with a pin hammer or heavy chisel, and they are easy to pull out.
You can leave them in if you wish but I never found them in original old work - I did loads of period windows including reclaiming the glass.
I also never used them in new work unless done in situ, where some support becomes necessary with bigger panes temporarily, but not permanently once the putty has gone off.
If left in they can become a problem and if sashes not maintained can rust and loosen putty.
Rusty panel pins are worse than sprigs to remove and I always associated them with badly fitted replacement glass, not original work.
But I hardly ever took glass out of any modern windows later than Edwardian, where big panes are common, but still without pins left in.
PS sprigs impossible to use with thin glazing bars such as you find in Victorian stuff, but completely unnecessary anyway with small lightweight panes.
PPS "Sprig hammers" seem to be a picture framers thing according to google. I'd never heard of them I'll have a look in Salaman.