If I run an IRC server, and my friend and I connect over TLS and chat, is the conversation essentially end-to-end encrypted because all computers that see the plaintext messages are owned by one of the participants?
No. End-to-end encryption means that the middle man cannot see anything. Even if you own the IRC server, it is not end-to-end encrypted. It's probably safer than using a public IRC server, but not as safe as using OTR or another client-side text encryption plugin that supports IRC.
The fact that the IRC server has access to your communications means it is, by definition, not end-to-end encrypted. Just because you trust the server doesn't change this.
No, IRC isn't end to end encrypted.
End to end encryption is not something you see usually. A few communication applications use it, because is something difficult to implement, and most use cases don't need it.
On IRC, the connection between the clients and the server are encrypted, but the server holds the keys to every connection. The server have access to the clear text of every single message.
The owner of the server have plaintext access to every single message sent on the server. Unless the clients themselves are encrypting the messages before sending, the owner have access to every single one of them.
It's the same with email, with FTP servers, with almost every online chat service...