Sorry, I didn't have a look to that before: there Mediaserver is selected.
In the last minutes MC crashed several times. Here are logging and system reports.
So if MediaServer is selected and is running you control MC with the menu bar icon.
Exiting the UI (if it's showing) with file->exit or by hitting the close button leaves media server running.
Closing via Apple Menu->Quit also terminates the media server instance.
Your log shows this at the end:
0051926: 6186250240: Playback: CCoreAudioPlugin::StartCoreAudio: Failed to start audio device (error code: 0x3c deviceId: 0x0 AudioDevProcID: 0x0 error: )
0051926: 6186250240: Playback: CCoreAudioPlugin::StartCoreAudio: Failed (second time) to start audio device (error code: 0x21646576 deviceId: 0x0 AudioDevProcID: 0x0 error: )
0051926: 6186250240: Playback: CCoreAudioPlugin::StartCoreAudio: Failed (caught handled exception)
0051926: 6186250240: Playback: CCoreAudioPlugin::ShutdownCoreAudio: Start
0051926: 6186250240: Playback: CCoreAudioPlugin::ShutdownCoreAudio: Releasing interfaces
0051926: 6186250240: Playback: CCoreAudioPlugin::ShutdownCoreAudio: Failed to release exclusive access
0051926: 6186250240: Playback: CCoreAudioPlugin::ShutdownCoreAudio: Original Bits: 32
0051926: 6186250240: Playback: CCoreAudioPlugin::ShutdownCoreAudio: Current Bits: 671174144
0051926: 6186250240: Playback: CCoreAudioPlugin::ShutdownCoreAudio: Failed to restore virtual format
0051926: 6186250240: Playback: CCoreAudioPlugin::ShutdownCoreAudio: Finish (0 ms)
0051926: 6186250240: Playback: CCoreAudioPlugin::StartCoreAudio: Finish (20513 ms)
0051926: 6186250240: Playback: CCoreAudioPlugin::Open: Finish (20513 ms)
An error in the audio device it seems. Looks to me like it's trying to open a non-existent device. I don't know why that causes a crash.