Stu's spot on there, WAV - there is no difference in register between your two voices, only one of tone &, to a certain extent diction, but even so it's hardly obvious until you point it out. There's certainly no falsetto (for an example of which see HERE).

If I might be frank here (speaking as one of the only ones here who has not only voiced any appreciation for your singing at all, but openly welcome your participation in singarounds) it seems your vocal intonation comes completely unstuck when you use accompaniment. Unaccompanied such things are rather less evident, and, to a certain extent, less important, but once you've got that keyboard in there it's only effect is to highlight your wayward intonation. Given both the modal quality of your chants and your fondness for E. trads, my advice would be to try singing against a drone - but practise first by singing scales, modes & intervals. Once you've got your 3rds, 4ths, 5ths and 8ves sorted you should find things a lot easier. Also, purely as a excercise, try singing whilst playing the recorder - something many flautists do as part of their sound (such as Ian Anderson from Jethro Tull (who nicked his entire flute act from Rahsaan Roland Kirk who pioneered such techniques). I'm not suggesting you make this part of your performance, but it might help improve your intonations generally, and get your voice in tune with your flute when you use it for preludes.