I have the AuthorizedDamageRequiresOwnership flag set, but players cannot damage their own twig.

Here is the other config:

"TwigDamage (FLAG)": {
"Apply To Twig (when TwigDamage flag is not set": false,

Am I missing something?

Those settings control allowing or blocking damage from other players, not damaging your own twig.

For players to damage their own entities, including twig, you need to use AuthorizedDamage and AuthorizedDamageRequiresOwnership flags. These are explained in the documentation too, and does not require any other flags or rules. TwigDamage flags are not related to these flags.

If it's still not working, another plugin is likely blocking it.