I think 'interject' has a sense of interrupt or breaking into (*ject -- inject; eject; project; reject; deject; traject;... all carry the same sense);

and 'interpolate' has a sense of 'joining' or two extremes (poles -- polar (adjective) -- *pol-ate (to make into a verb))

Interface is about a 'place' between independent/different 'entities';

Interact is about (yes) act[ion] between entities.

;-)