More specifically, how does one know or how does one check if a cardano-node process is acting/functioning as a relay versus a producer?
It depends on how you configure your nodes, there are few things that changes between a block producer and a relay node. In the block producer's topology.json you should only have your 1/2 "private" relays. Another thing that you can check is the start up script of the node, the block producer need to include different keys: kes, vrf, ecc inside of it. The last thing that comes to my mind is that if everything is configured in the corrected way, you should be able to see different things by opening the liveview of the node, first of all, it tells you if you are running a core or a relay node. If you are watching the core node, you should be able to see processed tx and mempool tx/bytes, plus the kes expiration date, if not, you should be able to see peers/other's pools relay nodes.