How to wire line out on a record player?
Whenever attaching an RCA plug (or any coaxial plug) to a shielded cable, always connect the center conductor of the cable to the center pin of the plug, and the shield of the cable to the outer shell of the plug. Other configurations are almost never what you want.

You do not need a separate ground wire because there is an internal pre-amp providing a low impedance output, so special configurations of ground wiring are not necessary — the cable shields are all the ground path needed.

You will need to figure out which channel is left and which is right if you want your RCA plugs properly color coded — I don't know of a good way to do that short of swapping them if you get it wrong, or tracing wires all the way back to the cartridge, or making a temporary alligator clip connection and listening to that. (A continuity test will not work by itself because of the pre-amp.)