OP, although, Bitcoin isn't technically a peer-to-peer network. It's a broadcast network where the transactions go to the mempool, then the miners pull the transactions from the mempool to include in a block. It's not really going from one peer directly to another peer.
Miners don't control your funds or even the transactions in the mempool, they simply use their computational power to mine blocks and get rewarded for it, the system isn't controlled by a central authority, it's all decentralized and happens within the ecosystem, which is why you can't say that Bitcoin transactions aren't P2P, because they are, since you send the funds, and I receive them directly without them going to a third-party before reaching me where the third-party can possibly stop them from reaching me or make any changes in it, but that's not the case here.
To @OP, I think we can simply say that Bitcoin is a decentralized peer to peer payment system, and P2P systems where escrows or third-parties are involved, such as the ones we see within centralized exchanges, they are centralized P2P systems, could also be called pseudo P2P, but we can't say they are fake P2P, it's just that both have different working systems. In a decentralized P2P system, the funds go directly from one peer to the other, and in a centralized P2P system, the funds first go to the third-party, and then they are released to the receiving peer upon confirmation from the sending peer.