by Ken Task.
Home connections to an ISP typically have some device on the edge of the home network that interfaces/talks to the upstream ISP.
Home Network -> device -> ISP
Typically, (but not always) upstream ISP allocates a public ip address to the device and probably uses DHCP to set the lease of that IP address. Lease is how long that device gets that public IP - could be forever until ISP re-cycles or some period of time - a year?.
That's what one is working with with noip or apps like that ... they inform about changes the IP address assigned to the outside interface of that device and changes DNS accordingly.
Example: yourserver.noip.com used to be XX.XX.XX.XX and is now YY.YY.YY.YY after re-cycling by upstream provider.
In config of moodle, best to use a resolvable fully qualified domain name.
Now you've mentioned 'port forwarding' ,,, that must be in your device that connects your home network to upstream provider. Or are you using your NAS for that ... has the capability of acting as your home network router and thus gateway to the internet. NIC 1 -> inside home network ... NIC 2 -> upstream provider.
If that is allowable by your ISP and setup correctly, in theory, what you are wanting to do should work ... but, as you said, could be further firewalls/network configs etc. in play that you cannot control.
Many of those devices (home router) have a web based interface that internal private IP addresses can access for configuration of the device. Some have testing tools ... ping for example ... not sure I've seen DNS checking tools, but if you have such a tool, and ping'd by the fully qualified domain name setup in noip or where ever you did that, the ping has to look up the FQDN first before ping knows where to go.
All this to say ... check the device that sits on your boundary of the home network.
Also, if that works, remember to keep whatever is being accessed by the public internet up to date ... that includes your Moodle.
Good luck!
'spirit of sharing', Ken