Thinks I hear people say and have no idea how the concept entered their heads:
-Revolvers are more powerful. They shoot the same bullets. Fire a 9mm round out of an M9. Now fire the same type of bullet out of some kind of revolver. There is zero differnece in stopping power. Yes, higher caliber revolvers are more common than higher caliber automatics, but both exist.
Technically, if same cartridge and bullet is used in a revolver and a semi-automatic pistol, and energy losses from the inferior gas seal on the revolver are ignored, then the bullet fired with pistol will have slightly lower muzzle velocity.
Why? Because some of the energy of the gas expansion will be directed to cycle the weapon and chamber a new round, instead of propelling the bullet all the way through the barrel.
It's the same as with bolt action and automatic rifles. However, the energy lost to the cycling mechanism isn't significant enough to say that the "power" of the gun would diminish in any meaningful way.
- All automatics have higher recoil. It depends on the caliber, size of barrel (length and width together), chamber seal, and the way the action cycles the weapon.
Expanding from the previous, comparing the same cartridge and round fired with revolver and semi-auto pistol, the revolver will have slightly more recoil simply because the bullet should theoretically achieve slightly higher muzzle velocity.
However, the matter is not as simple as that. Part of the recoil comes from the combustion gases shooting forward from the muzzle after the round, and that part of the recoil can be reduced with different muzzle brake configurations. A good muzzle brake can make a big difference in the amount of overall recoil a weapon produces, even though the muzzle velocity and mass of the projectile stay constant.
Mostly, though, you are correct - there's no distinct difference in recoil between one weapon type and another, if projectile mass and muzzle velocity are the same. Bigger differences come from the geometry of individual weapons, which cause secondary recoil effects like muzzle climb.