Beyond that, there are cases where you absolutely are referring to a specific one, but use a/an.
"Our house has a pool" for example. There is a very specific pool being referred to, but it's sufficiently indirect that "the" would never be used here without radically changing sentence meaning.
The reason you say "our house has a pool" is not that you're not referring to a specific pool. It's that the pool you're referring to is not already present in the conversational context, and it needs to be introduced. A(n) performs that introduction.
If you refer to that same pool without introducing it, it will still be marked for definiteness - compare my pool, my house's pool, or the pool at my house. But your example sentence doesn't work that way.
(Note that the three phrases I listed are all bare noun phrases that can't stand as sentences. Note also that if you used one of those phrases when your listener wasn't already aware that you had a pool, they might feel that you "moved too fast" in the conversation, assuming that they knew about your pool when they didn't.)
You can use "the" in the original sentence just fine if the pool had been established earlier, like with planning a pool party. Then the pool is the pool that you'd be using, and not simply a pool that will be found at some point in the future.
"Our house has a pool" for example. There is a very specific pool being referred to, but it's sufficiently indirect that "the" would never be used here without radically changing sentence meaning.