> I read that Atatürk chose his as "Father/Sire/Source of the Turks"

IIRC, the Parliament (Grand National Assembly of Turkey) gave Atatürk his "surname".

It might give more context to add his definition of "Turk" is similar to "American". Basically, it's about how you feel; if you feel like you're Turkish, you're Turkish. It's not tied to a specific race.