It all comes down to perceived recoil. A well fitted gun with a stock of proper length and fitted will kick a lot less than poor stock fit. Even minor changes to angle of butt plate makes a big difference.
Then comes technique. A proper hold with the forehand on the barrels with a straightish arm acts like a spring soaking up the recoil. A bent forehand gripping the foreend close to the action soaks up very little recoil.
The weight of shot. A heavier load will generate more recoil than a lighter load. A faster load will also generate more - laws of physics.
Gun weight also helps - a heavier gun kicks less with a slower recoil than a lighter gun.
But barrel diameter also has a big difference. A 28gram load in a 12 will require less pressure to achieve required velocity than a 28gram load in a 20. And back bored / over bored barrels recoil less. Take a 7lb 12 and a 7lb 20 both shooting a 28 gram load and the 20 will have a snappier kick from the higher pressure.
Then it comes down to the gun. A well made gun put together will feel much much smoother to shoot than a cheaper poorly made gun. I have shot pump guns - they are generally mass made and quite ratly. They kick. Semi’s do soak up some of the kick but you often feel a multiple kick. And they, like pumps tend to vibrate. And most pumps require a 28gr load to run reliably.
If you want a nice smooth shooting gun with minimal recoil I would find a good well fitting 12 bore over and under and use 24gram loads. Cost need not be high. Things like the ATAs are very good value for money and will be much nicer to shoot than similarly priced semis.
And don’t be put off by lighter loads on game. Drop down a shot size to keep pattern density, ignore those birds where it’s more hope than expectation and instead concentrate on birds you can kill and kill cleanly. By picking your shots your kill rate goes up and so does confidence.