I think you are confusing MIDI with audio. MIDI is not the sounds from your Korg, MIDI is just data that tells the korg synth what to output to audio.
in order to record a specific path on your korg in MC, you can do that a couple of ways. you can connect audio cable(s) from the korg to your soundcard, set up an audio track in MC, hit record and play the korg.
or if you want the flexibility of being able to edit the MIDI so that your performance is a little more close to perfect than it would be just by playing on your own, you can record the MIDI on a MIDI track of your playing. don't worry about the korg patch at this time. just output your korg to a monitor or something so you can hear yourself play.
then go edit your MIDI to your hearts' content.
then you'll need to set up an output MIDI cable from your soundcard (the input will work, you won't need it for this step), and again, set up the audio cables to go to your soundcard.
set up an audio track in MC to record what comes out of the korg, set up your MIDI track to output to the korg and set it up to the correct bank/patch you want to record.
hit record in MC. the MIDI will be sent to the korg via the MIDI cable (could be usb) which sends the audio to the cables to the soundcard and records the audio in the audio track in MC.