Does this mean, "I do not speak on his behalf"?
yes. same in albanian.
"... in his name" or "... on his behalf", never "... in his behalf"
Could you also say, "I do not speak for him" or would that be too vague?
I think that should be accepted, but I'm not a native speaker.
This doesnt translate correctly
It may well not be an expression you're familiar with, but "I do not speak in his name" has a clear and particular meaning: I am not his agent; I am not speaking on his behalf.
This would be a fairly formal thing to say, but in business or diplomacy could be useful.