I agree with Cool Beans that song books and more often the internet are often wrong, but Woody did often play an Ab chord. He did this by applying a capo (Woody called it a "clamper") at the first fret and fingering a G chord.

As Leadfingers implied, there are more than one correct way to chord a song. Because we don't play the same chords for a particular song doesn't mean that one of us is wrong (unless we try playing them at the same time).

Playing Will The Circle. . ., some folks throw a VI minor chord in right before "in the sky" and some leave it out. Both are right.

Some folks like to play Tenessee Waltz as:

G |G |G |C |G |G |D7 |D7 |. . .
While some get a bit fancier and play:

G |G |G |C |G |E7 |A7 |D7 |. . .
James Taylor played Oh Susannah simply as an instrumental, but when he backed up his vocals, he added a lot of chord substitutions.
None of these chord progressions are wrong. They're just different.