Thanks to Martin Ryan's information, I was able to find another version, which I think is better than the one in the Lincoln quote:

Copied from http://freepages.genealogy.rootsweb.com/~irelandlist/stpat.html

THE BIRTH OF SAINT PATRICK
By Samuel Lover
b. Dublin Ireland 1797 - d. 1868

On the eighth day of March it was, some people say,
That Saint Pathrick at midnight he first saw the day;
While others declare 'twas the ninth he was born,
And 'twas all a mistake between midnight and morn;
For mistakes will occur in a hurry and shock,
And some blamed the babby - and some blamed the clock -
'Till with all their cross questions sure no one could know,
If the child was too fast - or the clock was too slow.

Now the first faction fight in owld Ireland, they say,
Was all on account of Saint Pathrick's birthday,
Some fought for the eighth - for the ninth more would die,
And who wouldn't see right, sure they blacken'd his eye!
At last, both the factions so positive grew,
That each kept a birthday, so Pat then had two,
'Till Father Mulcahy, who show'd them their sins,
Said "No one could have two birthdays, but a twins."

Says he, "Boys, don't be fightin' for eight or for nine,
Don't be always dividin' - but sometimes combine;
Combine eight with nine, and seventeen is the mark,
So let that be his birthday." - "Amen," says the clerk.
"If he wasn't a twins, sure our hist'ry will show
That, at least, he's worth any two saints that we know!"
Then they all got blind dhrunk - which complated their bliss,
And we keep up the practice from that day to this.

[Sheet music is included in a book called "Irish Music" or "The Library of Irish Music."]