What is the difference between waveclear and pushing power?

In the context of the answer you linked, 'waveclear' is defensive, 'pushing' is offensive. A champ is said to have good defensive waveclear if they sit under their turret, quickly clear out a minion wave, and NOT EXPOSED THEMSELVES TO DANGER WHILE DOING SO. I suspect that last part is what was being referred to in the comment you linked. Since Trynd is a melee champ, he has to go out and meet a wave to kill it, which exposes him to enemy ranged champs who are seiging the turret.

In general, I've also heard 'waveclear' used to just refer to killing a minion wave quickly (usually through AoE or many-target abilities), without any implication of whether it was defensive or offensive. 'Pushing' always refers to the offensive capability of being able to escort a large amount your minion wave to an enemy turret and put damage on the turret. I'm unaware of any term that emphesizes purely defensive 'waveclear', which I suspect is why the author of the comment you linked just used 'waveclear'.

A champion that is said to have good waveclear usually have a skill (passive or not) that deals damage to multiple targets, making clearing a wave or merged waves easy and fast.

A champion that is said to be able to push is a champion that usually hit hard and can take down a tower fast.

A champion can push by also making sure a lot of their minions hit the enemy's tower.

The example given in the second comment of this answer about Tryndamere is a good example of a champion with pushing power but not good waveclear.

Caitlyn is a champion with good waveclear and can, with proper items, get good pushing powers.

Yi usually ends having both for example, with alpha strike to clear and hitting hard to push.

The other answers are good, to add one more thought to the mix though.

Waveclear is about damage to minions only. Champions with lots of area damage are excellent at wave clear. Pushing also involves damage to turrets.

Ziggs/Brand/Morgana/etc will clear a group of minions very fast but it will take them an hour to kill the turret as that can only be done with auto attacks.

Yi will clear the minions fast and destroy the turret in seconds, but he has to be in melee to do it.

Jax will take a while to kill the minions (he has to hit each one unless he has an unusual item choice) as will quite a few Nasus builds - but once they get to the turret they will take it down very fast.

Caitlyn and Twisted Fate (because TF builds Lich Bane) will both clear waves fast and destroy turrets fast and be able to do both from a safer longer ranged position.

Pushing is how a wave is moving. A champion can push a wave by helping the winions kill enemy minions so the winions have the strength advantage.

Pushing power is the ability of a champion to quickly help winions push into objectives.

Wave clear is simply the ability to clear a wave, it can be both offensive and defensive.

Offensive wave clear: clearing a wave with the intention of shifting which way it is pushing to benefit your team.

Defensive wave clear: clearing a wave to avoid it pushing against you. One might do this to avoid allowing the enemy team to get shots on towers while the minions would be tanking.

A game without champion involvement will have both pushing and waveclear. I really don't think these terms have to be champion specific.