The 600:5 and similar ratings assume that the relay or other device connected to the low current side of the CT is designed to operate with a 5A full span current.
That is why it says 600:5 and not just 120:1.
If your device wants to see a full scale current other than 5A, then you have to do a little more work in choosing the CT.
You can use a 600:5 CT with a device that wants to see 1A when the primary current is 120A. It will work just fine, but will be overbuilt for that task.
If you want to get a 10A output to your device, you could not use any of the CTs listed because they would be seriously saturated before that point.
None of these ratings directly address fault current. Using a CT instead of a suitable shunt for thermal trip and coil for magnetic trip is an entirely different design problem.