Does Storm count opponents' spells?
You get a copy for every previous spell, not just the ones you cast. The Storm rules (that you linked to) define Storm in rule 702.39a:


  Storm is a triggered ability that functions on the stack. “Storm” means “When you cast this spell, put a copy of it onto the stack for each other spell that was cast before it this turn. If the spell has any targets, you may choose new targets for any of the copies.”


As you can see, it uses the same wording as the reminder text you quote. It does not mention who cast the previous spells, so it counts all previous spells.

Cards that care about which player did something always say so. If any trigger event, or description, or bit of text that cares about information does not mention a player, then it does not distinguish between players.