The discordant variant (Bb D F) is what I was taught, being taught by hearing and no written music. As far as which is "correct", that is harder to say.

I think that having the D there makes it easier to sing, melody-wise, but the C works better harmonically. This probably makes the C variant correct, at least from a theoretical perspective where you want decent harmony throughout the whole of the singing. But if pilgrims not extremely skilled in singing were messing up the Bb C F, they might have just started singing Bb D F, not minding the dissonance much so long as they could continue singing on the right note once they reached the G. And that variant may have become more popular, because when people are learning a round, they must first learn (and then first remember when they mentally dig it up later) the melody, the harmony results secondarily, from a temporal perspective. So you're most likely to remember a melody which is more intuitive.

That's my uninformed two cents.

The pdf is there in your original post, but I find that I can't see attachments when on my phone, so perhaps something similar is why you can't see it now.