They are not the same.
The Miller 14 pin provides 24V &115 V, remote amperage with a 1K ohm pot and remote contactor.
The Lincoln 14 pin provides 42V & 115V, remote amperage with a 10K pot, and remote contactor.
Someone who understands what needs to be changed could alter the 12rc to operate off the Lincoln, but there is more to it than just changing to a compatible plug.
Specificly the 12rc operates on 24VAC and uses the 1K ohm pot for voltage control. You need to add a transformer to provide 24 volts from the 115, and change to a 10K ohm pot for voltage control.