A more interesting definition is that a server is something that is asymmetric from a client. Things that are peer to peer which are symmetric to each other are something special.
While you can frame peer-to-peer interactions as as a series of server-client interactions with servers and clients swapping roles... you shouldn't. It's a different paradigm with different implications.
Individual transactions might be usefully described as server-client, but the overall system is not server based. So...serverless.
In all cases, one node must send the first packet. The node sending the packet is the client, and the destination of the packet is the server.
Also, the internet is a peer-to-peer system. But "peer-to-peer" is an abstract paradigm, because ultimately, (a) two peers need to know about each other, and (b) one peer needs to initiate every transaction.
I realize we're just arguing semantics here, and I'm not sure what point I'm trying to make, but it's an interesting discussion nonetheless...
> In all cases, one node must send the first packet. The node sending the packet is the client, and the destination of the packet is the server.
I'm pretty sure you are just making up new definitions. It's kinda like saying that whoever says the first word in class is the teacher and the rest are student.
The fact is that both peers send packets and the first packet isn't particularly significant, except from a stateful connection or firewall perspective.
Serverless as 3) An application which run on a network and communicate with others on an equal status, you cannot distinguish one node from the others based on the source code or the logic it is running. In a sense even amazon serverless platform is not server less but it is server lite.