The statement below is false.
The statement above is correct.
The text is:
At the start of a battle:
If you are the defender and have a Goliath in the battle, your opponent must destroy 1 of his participating flying units.
This can indeed be read both ways - to me, the one saying "each Goliath kills a flying unit" makes a bit more sense, but maybe we should make it a rules question to the FFG guys. Or does anybody have more conclusive evidence either way?
Edit: It only says "a Goliath" and not "at least one Goliath", so I disagree with Smuggler. The case isn't clear at all. "One Goliath" wouldn't make it any clearer either way, either.