how do I backup my iPhone 13 Pro Max to an external desktop hard drive?

This cannot be done, directly.

You'd have to backup your iPhone to your computer using iTunes, then the computer must complete a backup of its own hard drive, to the external hard drive.

There is no supported way to back up to anything other than a specific location on the system drive. This is partly to protect the backup from viewing by a hacker who has hacked the computer. If you do a web search you will find several articles on how to do what you want, but be aware you are giving up some privacy by doing so.