I'll classify pusher as someone who can easily bring up a huge horde of minions to the tower easily a la Sivir/old Alistar (he can't even do that nowadays due to his heal nerf).
While I don't think it's the funnest play style, I do miss them quite a bit.

The last one just got nerfed into the ground. I don't think Riot is promoting that play style.

Heimer isn't exactly "lately" though. He's..something like 10 champs ago? The rest had all been a tank-fighter hybrid, with a few exception of tank-mage hybrid.

I'd say Galio promotes pushing pretty well. His tornado even speeds up creeps.

Haven't played Kennen, Galio, Urgot?...3 out of the last 6 or 7 champs. I'm not saying that they are true pushers but they can push very well. Kennen erases minions. Galio and Urgot take a bit longer to be pushers but can do so very effectively mid game while harassing effectively early game.

Vlad pushes and farms like a boss.

Morde is the king off farming/pushing

Malz is a great farmer, but he was released like...2 or 3 months ago.