Hi Joe,
The earliest version I have is 'The Courting Coat' printed on the 1st December 1856 by Glasgow Poet's Box. Probably the direct source of the Grieg-Duncan versions. There is no stanza about the child being born with the ...... on.

Pit boots, worn by miners
Navvy boots, worn by navvies
Kettle Smock, as worn by farm hands in earlier centuries, a one-piece smock sometimes decorated to show ones trade, shepherd, ploughman etc.
Courting Coat just suggests a smart coat to attract the opposite sex.

Courting Coat

Toiling and shearing as I went on my way,
'Twas the thought of my darling led my mind astray,
The moon shone the light as I trudged along,
And I rode away with my courting coat on.

For courtship this night i was fully prepared,
I spat on my staff and I shaved my beard,
The moon shone more light as I trudged along,
And I rode away with my courting coat on.

I came to my love's window, crying ae you in bed?
As soon as she heard me she lifted her head;
She lifted her head, and she cries, is that John?
Oh, yes it is I, with my courting coat on.

She opened the door and guided me in,
I sat doon by the fire to warm my shoon (shins?)
The room door being open, the blankets fell down,
He jumped into bed with his courting coat on. (I/my)

She began to chastise me for being so bold,
I'm sure it's no harm to get you with child;
For all that I've done I count it but fun,
And I rode away with my courting coat on.

Come all ye fair maids, take a warning by me,
Beware of the laddie that's single and free,
For his heart it is light and his mind it is YON--
Beware of the lad with the courting coat on.

This version printed on Dec 30th, 1876.(repeated from 1856)

The kettle smock versions come from the southern counties c1900 and the Navvy versions somewhat later in the 30s. The pit boot versions may be a product of the revival, one of Bert's specials in all likelihood.