Peer to Peer networks are any network where
all participants in that network are considered
peers aka
equals and there is no hierarchy.
What does it mean when a transaction or exchange is truely peer-to-peer?
There is a slight difference here though.
A transaction (like a bitcoin tx) is 100% peer-to-peer since you are not leaving the network and you do not need something else. You send bitcoin and the other party receives it.
In a trade, you are getting something else that does not belong to this P2P network involved. For example you are exchanging your bitcoins for fiat or bitcoin for another cryptocurrency. That additional thing sometimes beings in centralization and lack of P2P-ness!
For example if you want to exchange your bitcoin for dollar, because of the centralized nature of fiat and reversibility of majority of fiat-related payment methods, sometimes you need that centralization too if you want to be safe.
In other cases it is not necessary even if it is used. Like swapping bitcoin for some other altcoin like LTC. It could be done with protocols like Atomic Swap and in a purely decentralized and P2P way without needing any third party. Which is what you said here yourself:
So, for an exchange between peers to be truely p2p and decentralized, it has to happen between exchanging peers who control their funds or private keys rather than one or more third party platforms holding the funds on their behalf. Both the sending and receiving peers have to be in charge of their private keys.