UPDATE:
Our conversation with our landlord went well. He brought up the new lease (seems standard but I'll read through it before we sign). I brought up our interest in buying a house without requesting anything in particular - just said that we've been looking on and off, and if we did find one, how might that play out with the lease. He nodded, but then said, well, what are you thinking?
Based on everyone's help above, I told him if that we found a house, we'd let him know once we were under contract, but that we wouldn't want to give notice until we closed. He definitely understood that! He initially suggested 3 months notice, but we offered 2 months, and he thought that would be okay. Said he wouldn't want to stand in our way of buying a house, though he did seem to hope that we wouldn't find one in the middle of winter.
He's going to type up an addendum and bring it up later. All in all, it went pretty well! My partner felt strongly about having a lease (and not just month-to-month), which is fine with me.
Our landlord requested additional money toward our security deposit to bring the deposit in line with our current monthly rent. I was glad he did NOT request a pet deposit or additional monthly pet fees, so that's excellent. We segued into talk about some minor repairs about the place, and that's all!
I appreciate everyone who weighed in - you gave me more confidence for this conversation.