MellinTransform yields DiractDelta function

I want to do a Mellin transform for the following function

psi[x_] := Sum[1 - E^(-x/2^h), {h, 0, Infinity}]


If I try MellinTransform, I get

MellinTransform[psi[x], x, s]
(* Sum[2*Pi*DiracDelta[I*s] - Gamma[s]/(2^(-h))^s, {h, 0, Infinity}] *)


But if I do it directly, I get

Sum[Integrate[(1 - E^(-x/2^h)) x^(s - 1), {x, 0, Infinity}, Assumptions -> -1 < Re[s] < 0], {h, 0, Infinity}]
(* Gamma[s]/(-1 + 2^s) *)


Which one is correct? Is there anyway to get the second form from the first one? How can we interpret DiracDelta[I*s] in the first output? What does something like DiracDelta[-I/2] mean?