A wonderful song!
The tune is indeed the same as "The Mist-Covered Mountains": however, when our group sings it, it's in A minor, so the F remains natural. Instrumental versions I have are the same notes, but with a G key sig, so the F gets sharpened. It makes for a very different sound.
I am a new member of this group, so I'm not too sure what their source was. Lyrics are the same as you listed, although you have an additional verse not in our version ("where was our highland mettle"), which I really like, so I think I'm going to see about adding it to our version!