The map also confirms that Mingulay is in the Hebridean Sea and not near either of the Minches.

It doesn't have to be right next door - it's the area they fish in. (and its quite possible that the writer of the song got his geography slightly out).

It says nothing about the origins of the names.

It doesn't? Whoops, I should have read the whole article! Not Gaelic? Um, I know it sounds silly (for western Scotland) but how about French? Their name for the English channel is Le Manche (the sleeve). Could some Gallic explorer have used it in the Hebrides as well?