Assuming that this is for a material handling magnet on a crane of some sort, I have a few observations based on working on locomotive cranes with large magnets used for handling scrap metal:
600 V insulation is likely to be fine. 2,000 V is a lot of voltage and it would be very unusual to find that anywhere where untrained people are likely to be able to get close to it. The old wire was probably either DLO (diesel locomotive cable) or Type W mining cable. The new wire is probably welding cable or maybe finely stranded UL Style 1284.
If the conductor (the copper part, not including the insulation) is 8 mm in diameter it is roughly equivalent to 1/0 AWG wire and will carry about 245 Amps, a 13 mm conductor is roughly 4/0 AWG with an ampacity of 380 Amps. That's a big difference, so it would be good to check the magnet for markings or trace the wire back to the source in the crane and see how the controller is rated / fused.
The wires supplying a magnet are subjected to very rough service and larger wires may be used just for their mechanical strength even if the size is not needed for the current to be carried. We used welding lead wire which has lots of small strands which made it very flexible and less likely to be broken by the flexing it encountered connecting a swinging magnet to the crane.
The wire run is likely to be pretty long (at least twice the length of the boom of the crane) so the size of the wire may have increased to manage the voltage drop due to the resistance of the wire. If you're running over 100' (~30 meters) then you probably want to step the wire up one size (e.g., if you need 1/0 for current carrying capacity, you'd go to 2/0 to manage the voltage drop).
Without knowing the details of the crane and magnet it is hard to be more specific, but hopefully this will help you to evaluate the situation and decide if the new cable is adequate.