linux - Should you use a private hostname or public hostname as server hostname?

Assume you have a web server and database server, the web server is publicly accessible, while the database is only used by the web server, their DNS is

web.example.com

web-internal.example.com

db-internal.example.com

So obviously the db should only use the internal DNS as the hostname, what about the web server? Should I also use the internal DNS as the hostname for the sake of consistency?