Box birds and pigeon shooting is one of my favorite subjects. On the question of what is a pigeon gun, aside from those ordered from the factory with no safety, if you can make money on box birds with a shotgun, then the gun you are shooting is a pigeon gun.
I started out with a Winchester M-23 12 ga, with 28" BBLs. Then went to a 682 Beretta International Trap. Later came a beautiful 32" barreled Perazzi Comp 1 with factory full chokes, an MX 8 Step Ribbed Perazzi, and finally an SC 3 Perazzi that was special ordered by an old friend who used to shoot pigeons for a living. I still have that one.
In between the Perazzi guns were a smattering of various SxS pigeon guns, mostly Parkers, just for fun.
Here's the SC 3
P-Gun R.jpg
Here's the trigger group that was set up by Allem's
Allem Release.jpg