Here's the way I learned it (Philadelphia, around 1962 or so):

I fell from a window, a second story window
And caught my eyelash on the sill, the sill
Go get an axe, there's a fly on baby's mustache
A boy's best friend is his mother
(I'm a moron, you're another)
Presents make the heart grow fonder
Peroxide makes the hair grow blonder
Christmas comes but once a season
My father shot a horse
Bang! Bang! Ne-e-e-i-i-i-gh!

That's how I learned it, though I don't know how it might have been taught to the person I learned it from. I do remember that my sister and I would laugh hysterically when we got to the line, "...there's a fly on baby's mustache." But those were simpler times.

Aloha,
Mark