With the Toto, we have to exert more pressure. Do you press the handle down slowly and release it? Or simply press down as quickly as possible and let it go?
Also, it makes this huge "gulping" sound before you hear the water flush. Is that normal?
The flapper in the new toilet is 3", not the 2" of the old toilet. It does take more effort to lift the flapper up with the force of the water in the tank holding it down. All my grandchildren are worthy of this though.
The gulping sound is the powerful flush as the water and solids are sucked through the trapway. You may need to find a new place to store your plunger as the odds are that it will become dusty with lack of use.