It's not technically a payment at that point, since the payments is atomic end to end. But yes, you send a message your peer, which sends it to another peer, which sends it to another peer...
like any other P2P network.
"A purely peer-to-peer version of electronic cash would allow online payments to be sent directly from one party to another without going through a financial institution." [1]
Furthermore, the previous title was "Electronic Cash Without a Trusted Third Party" [2]. So reading "Peer-to-Peer" as "Person-to-Person" would mean that the title hasn't changed in meaning, it just became a bit more catchy.
Also, when analysing the incentives of participants in the bitcoin network, it turns out that network nodes do not actually form the ideal-typical p2p mesh network (where all nodes are equally distributed and connect to a few other nodes) but a more densely connected network where connectivity to mining nodes is strongly incentivised. This topic has been researched and discussed by Dr. Wright (See [3] for more information).
[1] https://bitco.in/bitcoin.pdf
[2] https://nakamotostudies.org/literature/ecash/
[3] https://nchain.com/en/blog/bitcoin-network-topology-small-wo...
Wrights comments on topology are technobabble and largely meaningless, so it's difficult to say something about them... however, to the extent that we can assign any meaning at all to them don't you notice that saying your transactions need to go through particular nodes sounds an awful lot like the property you're using to argue that lightning is not peer to peer?