As a next step in understanding your situation, do the following;
- Reconnect your MIDI In cable.
- Record a brief passage as before but do not delete it.
- Play back the MIDI region. Does it sound ok or not?
- With the MIDI region selected, open the Event list. Is the note data only from MIDI channel 1 or is data from another channel or controller data also present?
Logic has a useful MIDI monitor. To show it, Control-click on the right side of the display as shown in the attachment, then select Custom. The MIDI monitor will not be visible on the right side of the display. You can see MIDI channel number, note name and controller data during input.
You'll need to have audio channels set up in Logic to hear the playback sound from Input 3-4, and to record the audio for your mix.
There may be other issues that are microkorg specific that I can't help you with.