I read that leveling weapon skills (such as one-handed and two handed) is not related not to the number of hits, but to the damage done (not counting power attacks).

But does it level faster by drinking a potion to increase damage ? And what about enchanting ? Say I have a piece of armor that increases one-handed damage by 40%, will it also level up 40% faster ?

I thought it was number of hits. I get many more skill ups fighting with my dagger then a sword, which results in more swings, less damage and longer fights.

If it is anything like the past games then it is by number of hits.

According to the TES wiki, the game mentions it's the number of hits that counts, but in reality it's the damage done.

Where did you read that it is by damage done?

I would of thought that it is similar to smithing, you can make a crappy iron dagger and still get the same experience as making a weapon of much higher quality.

I originally thought it was by damage done, but then considered how I frequently stealth kill with a dagger - with 15x damage, my one-handed skill would be maxed out by now.

I have a gaming guide that lists the details about every 1H/2H/Staff/weapon in the game, damage it does, weight, value. And the same for all the armor types. Unenchanted.

Unfortunately, it doesn't include anything about how damage is calculated.

I'd have thought that to be a very important piece of info for our calculating minds. Also looked at different wikis, etc., but nothing.

http://elderscrolls.wikia.com/wiki/One-Handed

It's damage done, as I learned beating on a troll with a dwarven sword, vs. iron dagger; one-handed leveled three times as fast with the sword. That said I don't think that critical and sneak damage contribute towards increasing the skill, although sneak attacks will level your stealth skill.

this is also true in the same way for alchemy, in that the bigger the value of potion created, the bigger the level up. Or at least it seems to me...

It seems like it's the number of hits to me... I use my ebony daggers a lot more than my daedric battleaxe... yet my two-handed skill is much higher than my one-handed. This is because every time I use my 2h I'm in a real fight where I land many blows, while I stealth kill everything in one hit with my daggers.