It seems as if the Mini connected to the original port is coming up as Fast Ethernet (FE) (100 Mbit/s) and then either not trying to negotiate Gigabit Ethernet (GbE) (1000 Mbit/s) or failing to do so.
The fact that the cable and port works on another computer could indicate that the other computer expects only FE or that it is more tolerant of failure during the negotiation.
If you want to pursue this, I would still try to lay my hands on a known good GbE cable, just for fault isolation.
I have zero chance of getting hold of that router, hence can only guess, but things to look for would be:
- does the router have one LED per ethernet port and the colour of the LED tells you whether you negotiated GbE or FE? (usually green for GbE and amber for FE or worse)
- if so, what does the LED say when connected to the working computer?
- does the router allow you to hard-code the ethernet settings and disable negotiation on a per-port basis? (if the router has this capability then you want not to be doing that unless you are connected to a very old device, and you know what you are doing, and you have to do this for it to work)
- there are even routers around where some ports are “special” i.e. not all ports are created equal - but you would need to read the manual that came with the router
Maybe you are happy that you got it working …