Is there a way to have the same domain point to a local IP and remote depending on network?

EDIT: Thanks for the info guys! Very excited to get this all set up

At the moment I have a bunch of self-hosting services hosted in the cloud. I plan to get rid of my cloud resources entirely and run stuff on some server hardware I acquired recently but my ISP doesn't give me a static IP and I'm behind a NAT or whatever it's called (the thing that makes multiple people's home connections be behind a single public IP) so I don't think I can even expose directly to the internet. So my plan is to have a very small and cheap server at a data center and proxy my actual server behind that.

My question is, is there a way that I can set things up so that the same domain can connect directly to the server when I'm at home, and to the proxy when I'm not? The difference would be what connection I'm connected to (my home WiFi vs 5G/others' WiFi). I'm thinking I could maybe run DNS on the server and configure my router to use that as a DNS server, but wouldn't my phone/laptop cache DNS entries? So it'd still try to connect to the local IP even when I'm out.