Hairpin NAT (or "NAT hairpinning") is the term for "...connecting from inside local network the router recognize that by connecting to public IP, he has to route it back onto some local IP address", and the Linux NAT implementation supports it.
For a homelab situation split-horizon DNS is just fine. You're going to have minimal duplication of records from the public DNS into the private DNS.
The canonical frustrating "bad split-horizon DNS" world I've seen, time and again, is a corporate network with a MSFT Active Directory named the same as the company's public-facing web property (e.g. "example.com" rather than "ad.example.com"). This creates the need to duplicate all the public-hosted resources into the internal "example.com" zone (and keep them updated when records inevitably change on the public Internet). It's make-work for no practical upside in that example.
For a homelab situation split-horizon DNS is just fine. You're going to have minimal duplication of records from the public DNS into the private DNS.
The canonical frustrating "bad split-horizon DNS" world I've seen, time and again, is a corporate network with a MSFT Active Directory named the same as the company's public-facing web property (e.g. "example.com" rather than "ad.example.com"). This creates the need to duplicate all the public-hosted resources into the internal "example.com" zone (and keep them updated when records inevitably change on the public Internet). It's make-work for no practical upside in that example.