Confirmed Spells:

Healing
Magical Ward
Bound Battleaxe
Chain Lightning
Circle of Protection
Clairvoyance
Fire Breath
Flames
Frost Bite
Frost Rune
Fire Ball
Ice Spike
Lightning Bolt
Turn Undead
Water Breathing
Some Kind of Fire AOE "Fire Bomb"?

Since spells like Fire ball and Fire breath or Frost Bite and Frost rune are separate , I'm assuming that there will not be alternative ways to cast each spell, they will be separate spells, but we will still be able to combine different spells to make new ones, and combine the same spell to make a stronger spell at the cost of more magicka

Fire will do the most damage
Ice damage will also drain Stamina
Lightning damage will also drain magicka.

source(s)?

I don't remember seeing Bound Axe or Water Breathing confirmed.. obviously they're going to be in anyways but where did you read that?

did you get this off the favourites list in the E3 video? anyways, fire breath is a shout. Not a spell.

Fire breath is a dragon shout not a magic spell. Frost rune is not a spell but a version of frost, just like the flame thrower is a version of fire, Different versions are attainable through method of execution, ie: tap button, hold button, etc....

I know for a fact water breathing is in, if you watch the g4tv e3 skyrim demo, when Todd goes over the skill screen, pause at alteration and read the skill description, one of them is Water Breathing.

Combine spells to make new ones? IIRC they confirmed they are working on it.

These are from the E3 15 minute demo, the favorites list and the skills menu, sorry if some of them are shouts i wasn't sure about all of them.
Under favorites in the demo, spells like frost rune, frost bite etc. were all separate spells.

Fire breath was a shout, as others have said. I haven't looked too closely at the favorites, but I can confirm that Bound Battleaxe and Frost Rune were in the favorites menu as separate spells, or at least I can confirm that they appear as such in the 15 minute demo. They didn't show any spell combinations in the video, nothing's been finalized/confirmed on that front. Waterbreathing popped up in the skils menu, as did paralyze (now in Alteration), and a couple other spells. FYI OP Circle of Protection is the new name for Turn Undead, based upon what Todd called it and the effect that we witnessed. The specific name "Turn Undead" hasn't been mentioned in relation to that spell, or anywhere else.

Edit: Also, origin reason I wanted to comment after seeing OP: your title is misleading. This info can all be derived from the demo. And I'd have to look again at the video, but I never noticed anything about having Ice Spike/Frost Bite and Chain Lightning/Lightning Bolt as separate spells. I remember seeing Frost Rune/Frost Bite both listed in the favorites, and we saw a lightning spell being used on Draugr (then a separate Storm Call shout).

I can't wait for ice spike!

I'm interested to see how these new "warding" spells will play out. I hope it isn't another "cast resistant percentage spell, now there is a glimmer on your character" sort of thing.

When they go through the bookmarks in the G4 demo you can see Bound Battleaxe in there.

Turn undead was mentioned as a specific spell in the skills menu under restoration, as was healing and magical ward.

And sorry for the title being a little misleading , the info was new... to me.

There's a Search function in the top-right. Yeah, it was difficult to notice the first ime for me too . Anyways you should use that anytime you're unsure of whether you're potentially doing a repost.

While it is not confirmed in any way the detail that those spells had separate listing on the favorites menu at least implies to some degree they are separate spells and not combined spells. If you just equipped frost spell and it could do each of those effects based on how you pressed the mouse button you would think it would be listed just as frost spell and not ice spike. But it easily just could be that ice spike has a smaller/different list of variations than we thought or something. Either way it really is just separate spells one way they disguise it one way they don't.

I wonder if they'll bring back some of those scrapped spell effects from Daggerfall like "Wall of Fire."