I have Omnath, Locus of Creation out. I play 3 lands, triggering his landfall ability 3 times. I gain 4 life, add some mana, deal some damage. I decide I want to gain some more life, get some more mana, and deal some more damage. So I return Omnath to my hand, then recast him. My understanding is that if I now play another land, Omnath's landfall will trigger and I will gain 4 life, because this Omnath has no memories of his previous triggers. Is this correct? Does bouncing or flickering Omnath reset his landfall ability?
Yes. Comprehensive rules 400.7:
An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence.
Followed by some exceptions, none of which apply here. As it's moved from the battlefield to your hand and then back to the battlefield again, it's a new object, so the next land entering the battlefield is the first one it's seen and you gain 4 life.