Ethernet is low voltage cable, so bonding and grounding is not necessary to meet any code requirements. However, you do want to protect the devices connected to the cable. Many folks here have chosen to use media converters (converting cat 6 to fiber) and fiber cabling between buildings because it nearly eliminates the risk of a lightning strike passing through the cable and damaging equipment on either end. YMMV.
I would also lay PVC conduit in the trench you dug, run the new cable in the conduit, and then lay warning tape on top of the conduit before you bury it. You want plenty of warning when someone starts digging near your cable.
If I understand your description, the STARLINK router is WiFi only and it does not have any Ethernet jacks. In my experience the lack of a hardwired ethernet jack is unusual for a home wifi router, but I've never worked with Starlink.
You have the right idea for the new connectivity, with the Ethernet cable between the STARLINK router and your new router in the shop. Your new router would be configured to run in AP mode, and it essentially just extends your current network to the shop. IP addresses and DNS will still be issued by the Starlink router. Most new routers will have a setup wizard that you run the first time, and one option will be to configure it in AP mode. You can always do that setup back at the house with the routers sitting side by side, using a short ethernet cable, and then move the new router to the barn once you have it configured.