Maybe the division is more apparent if you're using containers, and/or have more of a social group to communicate support levels to? Personally I'm just using NixOS and am still at the stage of "I'm going to reboot the router now, it might take ten minutes or it might take days" [0]
If we don't need this distinction, then perhaps "homebase" could actually make more sense than "homeprod" ? Slightly different connotations, less putting on a show for lots of people but more homey like for a small family?
[0] from a technical perspective I'd love to separate the router out onto its own machine, but I want fast wireguard/openvpn/TOR/etc so it's a VM on a larger machine and I'm not really looking to expand my power consumption footprint to another large machine. I suppose I could split it out to two routers, one with straight NAT+wireguard and one with openvpn/TOR/etc, then the main could run on its own low-power box.