Can this sentence only mean that "he" went to a different place to live or can it also mean that "he" went away from "me" at that moment (by walking) and not in the sense that he moved to a new place.
This is a great question!
Although the far more common meaning of "move away" is
[go] to a different place to live
It also can be used in the second way you listed
"he" went away from "me" at that moment (by walking)
Both are completely correct although the first is more common.
The word "move" is actually very versatile in English. For further explanations you can read this dictionary entry.
It can definitely mean either, provided you provide the right context. A sentence establishing that you and him are in the same place is enough.
We were walking together, but then he moved away from me.
Likewise, by establishing that you lived in the same area at a time, you can give the sentence the other definition.
We used to live in the same neighbourhood, but then he moved away from me.