Any pump shotgun is legal, provided the magazine does not hold more then 5 rounds; pistol grips (but not collapsible stocks) are legal. Semi-auto shotguns are also legal, with the same 5-round imitation. In addition, a semi-auto must have a standard stock (pistol grips or collapsible stocks are not permitted).
Stay away from a SxS for HD, especially a Stoeger. All else being equal, why limit yourself to two rounds (even in NYC), when you could have a 5+1 available? At zero-dark-thirty, under stress, you don't want to have to break open a SxS to reload after two rounds. The Stoeger guns are even more problematic, since they lack extractors/ejectors.
I live in NYC and own an 870 with a Blackhawk SpecOps adjustable, recoil reducing, pistol grip stock, and a Benelli M4 with a standard (non-pistol grip) stock. My ammo of choice had been Federal LE132 00 (00 buck, 9-pellets, FliteControl wad, lower velocity), but is now the new LE132 1B (#1 buck, 15-pellets, FliteControl wad, lower velocity).