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.

Does the introduction of an intermediary really solve the problem? The issue of trust was not fully eliminated here after introducing an escrow. What happened is that, the trust was shifted from one party to the transaction, to another person being the escrow. Now both parties will have to trust the intermediary to be honest enough to release funds to the right person when needed.

Escrow or not, just do your findings well before transacting p2p with anyone. Be sure that you are not dealing with a scammer 

Peer to Peer transaction either by buying or selling bitcoin which ever way has to be done among two genuine persons who wish to send Bitcoin directly to one another account not through the bank or any other third party .

It directly send funds from your digital wallet address to another with a decentralized global network which the miners will now verify your transactions successfully and records it  using cryptographic measures which records it on the block chain permanently without the involvement of any intermediary.

Ser, where in my post said that miners control your funds and those transactions in the mempool?

I'm not trying to offend you, BUT I believe you need to learn some basic understanding about the network.

   ¯\_(ツ)_/¯

Well said.
Once I was done reading the OP, what immediately came to mind was is there a distinction? And which type was being referred to in the Whitepaper by Satoshi?
Since transacting bitcoin became banned in my country, we turned to using the P2P feature of some crypto exchanges. In this case it involved a third party which is the crypto exchange. I have also done P2P without a third party, how for the sake of trust and reliability that the other party will keep their end of the transaction, we most rely on P2P with a third party when it involves strangers. But then like you said, I don't know if this is the context the whitepaper was referring to when it was being published.

To the buyer you meant to say here.
One simple way of interpreting this is simple;

Seller: Bitcoin is locked in escrow by the seller
Buyer: The buyer sends bitcoin to the seller
Seller: Payment is received from the buyer and confirmed
Escrow: Buyer receives the bitcoin from escrow


There is another reason for reliance on centralized exchange. Not all direct p2p market place is fully working as a decentralized exchange. How the decentralized escrow mechanism of a DEX is being designed is worthy of being considered when choosing a DEX exchange to use. Eliminating escrow service means accepting likely counterparty risk and DEX doesn’t mean it is safe.

Why does it matter? Exchanges have been the only way to break the barrier and initiate a trustless trade (especially for transactions that involve a small amount), so the double spending isn't a problem at all. If you're just trying to discuss the newbies about the differences, make sure to include the risk involved---- in decentralized p2p, you're trading money with someone you barely even know and there's no dispute arbitration team to freeze or make an immediate recovery should things go south.

Why the hassle when I can simply get connected with whomever, with a temporary hash and  API, and the transaction goes smoothly even if I decide to not comply after receiving my payment.

There are many Bitcoin nodes and Bitcoin mempools with their own configurations.

A transaction with very low fee rate can be not chosen by a mempool because of that node and mempool settings like dropping after 2016 blocks (about 14 days) but is still available in another or other mempools. If that low-fee-rate transaction is not yet dropped out of all Bitcoin mempools, it will stay there till getting a confirmation in the future or the user bumps the fee or double spends.