Is there a difference between overrun and overshoot landings or are they exactly the same, talking about a runway excursion?
They are more or less the same and are often used interchangeably. FAA AC 91-79 Runway Overrun Prevention defines runway overrun as:
A runway overrun event is a departure of the aircraft from the end of the intended landing runway surface.
It means that the aircraft has touched down on the runway before going off it. Overshoot is used in the same sense (note that overrun/overshoot occurs both in TO/landing).
However, there is one case where they are used differently. Overshooting the runway also can mean that the aircraft has touched beyond the end of the runway i.e. missed the runway entirely.