Matrix homeservers have two important domains. The user-visible server_name and the homeserver domain which serves most of the traffic but is really seen by users. The docs around this variable said "This is used by remote servers to connect to this server" which is very confusing because most of the remote server traffic actually goes the server domain, not the server_name domain. (The server_name domain is only used to fetch the .well-known file that points at the server domain).
I largely copied the wording from https://matrix-org.github.io/synapse/latest/usage/configuration/homeserver_sample_config.html as I found it much more clear.
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