It depends on the context, but with verbs of motion like ran or flew or went, "verb over something" almost always means "moved from one side to the other."
In the example with the deer, the only reasonable interpretation is that the deer crossed the hill. The deer couldn't have been running above the hill (floating in the air!) and if we just mean that the deer was running on top of the hill, we would probably say "on top".
The helicopter is a little different, because it was probably in the air above the hill, but again, verbs of motion with over usually imply "from one side to another". If you said "the helicopter hovered over the hill", that would obviously imply that it was standing still above the hill. Some context is always necessary, because helicopters can be flying without moving.
I think a good related thing to be aware of is the difference between over, above, and on top (of). All of them mean "higher than something else", but they have some subtle differences:
- over : covering or moving from one side to another. I wear a mask over my face; the bird flies over the hill.
- above : higher than but not touching. The helicopter hovers above the hill.
- on top (of) : touching the top of something. The hermit sleeps on top of the hill - if he slept over the hill it would mean he was on the other side; if he slept above the hill it would mean he was somehow higher than it but not on it.