The answer is written on the blocks. The forward euler integral transfer function is Ts/(z-1). So if you want to create your own integrator, you should have the same transfer function.
The lower part in your picture has the transfer function (neglecting saturation);
-0.12*K*Ts/(z-1)
The upper part in the picture has the transfer function (neglecting saturation);
-0.12*z/(z+0.12)
First you are forgetting about the time step, Ts. Second what does 0.12 do inside the loop in the upper part? Second, you might be thinking of backward euler integration since you have the delay in the feedback not feedforward part.
