All charging is done with DC internally. Level 1 and 2 charging uses a rectifier inside the car to convert AC to DC. Fast charging simply bypasses the rectifier.

The reason Level 2 charging is current-limited by your car -- even if you had a very high-current AC source available -- is that to take advantage of a high-current AC source your car would have to carry around a bigger, heavier rectifier. Which would decrease your EV's efficiency just by virtue of being big and heavy.