I am trying to use this for a specific Moodle server. Is it possible to use let’s encrypt service on an example URL like moodle.mywebsite.com?
The instruction shows I can use it for the entire domain, mywebsite.com. But, the instruction does not show for a specific server. My server is part of a college and I cannot use the entire domain name. The network department arranges that.
Yes, you can issue a Let’s Encrypt certificate for any public DNS name that you control, including a subdomain, including in cases where you don’t control the base domain. This is a common use case and it should work fine.