The guide is correct for extraordinary abilities:
This spell functions like polymorph,
This spell functions like alter self,
You keep all extraordinary special attacks and qualities derived from class levels, but you lose any from your normal form that are not derived from class levels.
Nothing in polymorph or shapechange changes this rule.
And you are correct for supernatural abilities, because of the shapechange rule you quote. I can’t find any official source that applies any caveats or qualifications to the quoted statement—for shapechange. Notably, alter self (and by extension, polymorph) has you keep your supernatural abilities (unless they depend on physical features you no longer have), but then alter self doesn’t give you the supernatural abilities of the new form. So it’s possible that the guide author either spoke too quickly, forgetting they were discussing both supernatural abilities, or else they got confused because the statement about keeping supernatural abilities would be true for alter self and polymorph.
It is also rather weird, really, to lose something like lay on hands or diamond body, so there might have been a bit of missing it just because they weren’t expecting it.
Anyway, because shapechange can swap forms as much as you want, a lot of times it doesn’t even really matter all that much. Also, Assume Supernatural Ability can cover the cases where it does.