"Mixo-" in ancient Greek means English "mixed".

I don't think anybody knows why the Greek theorists labelled their highest mode as "half Lydian". The mediaeval meaning was different anyway.

As I used it, the word was in the scheme for representing gapped modes I got from Campbell and Collinson (though they didn't invent it). The implication for the tune leeneia referred to is that despite being notated with tonal centre F and a key signature of two flats, there's no major third (the A might as well be flat) and no mixolydian seventh (the E might as well be natural), which together makes it unharmonizable in any conventional triadic chord pattern - I and V are both incomplete. There is an English morris dance tune with the same structure, but other than that the nearest parallel song I could find was from Mongolia. Kodaly seems to imply there are other examples from the Uralic-speaking peoples of the Volga but doesn't actually quote one in anything I've read.