Exactly, Seamus, but the McPeake tune is much more singable (in my opinion), hence its popularity. Hamilton was a writer as well as a publisher and contributed to Johnson's Scots Musical Museum. His Braes o' Bowhether (to the tune The Braes o' Balquhidder, just to confuse the issue!) was published at least fourteen years before Tannahill's song. Tannahill chose a different air than Hamilton/Burns although it seems to have disappeared in Scotland but strong in Ireland and America.
For comparison, here's Hamilton's song.

The Braes o' Bowhether.

Now the day's growin' lang lass,
an' sweet shines the weather,
an' we'll owre a' the hills,
to the Braes o' Bowhether.
Amang the Glens an' Rashy dens,
I'll prize thee without measure,
Within my arms, wi' a' thy charms,
I'll clasp my lovely treasure,
In sweetest Love, our time will move,
wi' mair than earthly pleasure;
By the little limpid streams,
On the Braes o' Bowhether.

An' I'll ay loe thee dearly,
Ilk day wes' forgather,
Syne we'll row on the fog,
By the Braes o' Bowhether;
To Pipe or Flute, when time will suit,
We'll dance like ony feather,
An', skip the knowes where Claver grows,
or stray amang the Heather;
Ay free frae strife in sic a life,
There, weary shall we never,
By the limpid little streams,
On the Braes o' Bowhether.