To be under the influence typically means to be intoxicated, or in this case, drunk.
Consider the statement "When she came back, I did tell her that her father had killed someone while driving while drunk and that he had been incarcerated.". Based on this "When she came back, I did tell her that her father had killed someone while driving under the influence and that he had been incarcerated." a true statement? Yes, no, or I don't know?