@Marlies, if you can ping the router with your laptop over this cable and can reach the Internet, then it's probably some other problem.

But did you ensure that the laptop isn't connected wirelessly to the router when testing the cable, e.g. falling back to a wireless connection to a main SSID? Can you ping the router from the laptop over the socket cable while the WiFi radio of the laptop is disabled?

You wrote the EAP works when connected to the switch directly, but not when connected to the socket. This indicates a problem with the cabling if you used the same switch port while testing.

Anyway, we would need much more information of your network setup to be able to help, »access to Internet« from the laptop is much too vague. Always check basic connectivity first and if the laptop/EAP can reach the controller system over port X if connected directly to the switch, but not so if the socket plugged into the very same switch port X, then the laptop almost certainly can't reach the router over this port X, too (except if you use a fancy VLAN setup or switch ACLs or a firewall).