When you have a question that concerns a keyword ability, it's best to lookup the definition of that keyword.
702.85a Cascade is a triggered ability that functions only while the spell with cascade is on the stack. “Cascade” means “When you cast this spell, exile cards from the top of your library until you exile a nonland card whose mana value is less than this spell’s mana value. You may cast that card without paying its mana cost if the resulting spell’s mana value is less than this spell’s mana value. Then put all cards exiled this way that weren’t cast on the bottom of your library in a random order.”
The part that's relevant is You may cast that card without paying its mana cost. Note that it permits you to take the cast action. This has a few implications because of Fires of Invention:
- You can't choose to cast it if it's not your turn.[CR 101.2][CR 608.2d]
- You can't choose to cast it if you've already cast two spells that turn.[CR 101.2][CR 608.2d]
- It would indeed count as a spell you cast it you choose to cast it. Fires of Invention counts the spells you cast, no matter how they were cast.
I'm trying to understand the confusion since both Fires of Invention and the reminder text for Cascade clearly say cast.
Perhaps it's because you didn't use the alternative cost provided by Fires of Invention to cast the card found using Cascade. That's not relevant.
Perhaps it's because it's an effect that allows you to cast the spell rather than the priority rules. That's not relevant.
Fires of Invention counts the spells you cast, no matter how they were cast.
101.2. When a rule or effect allows or directs something to happen, and another effect states that it can’t happen, the “can’t” effect takes precedence.
Example: If one effect reads “You may play an additional land this turn” and another reads “You can’t play lands this turn,” the effect that precludes you from playing lands wins.
608.2d [...] The player can’t choose an option that’s illegal or impossible, with the exception that having a library with no cards in it doesn’t make drawing a card an impossible action (see rule 121.3). [...]