I've just been thrown into this keep and have to get up to the top to escape!

Battlespire?

Why were you thrown into the keep? How will you escape once you reach the top?

Same here. And I enjoyed the second half of the game more than the first.
But with the Special Edition Capcom promises to add more characters, including my all-time favourite, Vergil. And since (as far as I understand) it's going to be the old Vergil, the true Vergil, I'm keeping that SE on my radar.
Toren?

No.
Your spirit got taken out of your body and dropped through a chasm into the deepest level, where you materialized with only your pants on.

No idea, haven't gotten there. :D
I've never actually played Toren, but the name does seem familiar.

Soul Reaver?

No. You are a human, and had a sword prior to being thrown in.

And you meet the villain who gloats at you and lets you go.

I don't think it does. It's just a piece of equipment you had and then don't have.

Instead you gain rocks to throw and then some daggers, until you shortly afterwards find other weapons. You also get to fight a shit-ton of ants.

Oh, and you can collect plenty of skulls, although I'm not sure if they're actually usable for anything.

Is it a platformer?

No. You can only move in four directions.

Is it an indie game? Is it part of a larger franchise?

No, but you're getting very close, LoG is said to be one of its spiritual successors in a way. :)
A very famous game that's not part of a series.

Another funny tidbit - one of the goblins has your diary! You get it about an hour in the game or so, and it's very useful to you - it's your menu, stat screen and map all in one! As well as place where you can read through and edit the name of you inventory items.

Stonekeep

It was a great game I played in school. Are you really playing it right now? It's so hardcore ;))