The article clearly defines someone who always lies to mean they only make false statements. It's not hard or ambiguous to determine various scenarios in which "all my hats are green" is false.

The solution article also discusses vacuous truths ("all my hats are green" is true if you have no hats, vacuously). Truth has a definition in formal logic too, but we've gotten far enough that this problem is already solvable and unambiguous.