How do Trainers work in Skyrim?

I've been to one or two trainers for Destruction magic already. Since then, if I go to another trainer (different skill) of a level that I've already trained at, the dialog says I've maxed out my training (5/5).

So, it seems to me that you can only train five times at a given training level (Common, Expert, Master) across all skills. However, given how many different skills there are in the game and how many Trainers there are, this seems rather crippling to me.

Can someone explain exactly how Training works? Is my above presumption correct?

You can only train 5 skill levels per (character) level.

So if you are level 10 and train destruction at a trainer you can train 5 times. Then you have to get to level 11 to be able to buy skill levels at a trainer.

That is the only restriction I know of (being lvl 62 atm).

Yes, you guessed right - but honestly, it is not a big matter. You should only use trainers for skills that you want to learn but don't want to USE routinely (for example you don't want to keep doing alchemy things all the time but want to abillity for better potions: use an alchemy trainer).

Because in skyrim you gain skill points by using the skill you level up your skills pretty fast just by using them constantly. So, if you wear heavy armor, use a one handed sword and shield, all those skills will grow rapidly without you needing a trainer.