A man of words and not of deeds
Is like a garden full of weeds
For when the weeds do begin to grow
Then doth the garden overflow.

This seems to be the origin of the first; found in a couple of books of quotations. Anon.
Also found in Nursery Rhyme Books, with this added to it:

It's like a lion at the door;
And when the door begins to crack,
It's like a stick across your back;
And when your back begins to smart,
It's like a penknife in your heart;
And when your heart begins to bleed,
You're dead, and dead, and dead, indeed.

Possibly a translation from Charles Perrault (1628-1705), "Contes de ma mere l'Oye," as "Mother Goose Tales." Someone may have the straight of this.

The entire rhyme, as you give it from the traditional music site, is found in Halliwell, 1846, "Nursery Rhymes of England," No. LXXV of "Fourth Class- Proverbs."
Halliwell remarked: "One version of the following song, which I believe to be the genuine one, is written on the last leaf of MS. Harl. 6580, between the lines of a fragment of an old charter, originally used for binding the book, in a hand of the end of the seventeenth century, but unfortunately it is scarcely adapted for the "ears polite" of modern days."

Gee, what are we missing??

The complete Halliwell in on line, and is worth downloading.

The Santee-ti rhyme obviously is a variant.