Joe's Detroit version- and the Chicago version posted by mrdux- are identical to the one in the Oxford Nursery Rhyme Book by the Opies'.

Should Becca and Willa go to the bottom of the class? No, others are permissible:
S. Baring Gould, 1895, "A Book of Nursery Songs and Rhymes," Methuen & Co.
X
Sing hey diddle diddle,
The cat and the fiddle,
The cow jumped over the moon,
The little dog laughed
To see such craft,
And the dish ran away with the spoon.

I haven't gone back to Halliwell (1846) but he may have another.

The oldest version known in print (1765) begins "High diddle diddle..." Some claim the verse originated at Bolton Abbey, Wharfdale. http://www.boltonabbey.com/highlights_diddle.htm
Hey diddle

"...to see such fun" also is widespread.
http://www.rhymes.org.uk/hey_diddle_diddle.htm