Hi wchzym,
in your second circuit, if the right terminal of R9b (or directly the non-inverting input) were at ground and the input voltage were applied to the right terminal of R8b (instead of grounding it) this would be a standard precision rectifier in inverting configuration.
Changing the role of ithe two inputs (right terminals of R9b and R8b) does not transform inverting in non-inverting without any other change.
In the first case, you have a virtual ground at the inverting input if the OA. In the second, you have a "virtual Uin" at the non-inverting input. As a consequence, you have non-null output from the circuit when input is negative.
I suggest: using the principle of virtual ground, analyse whay happens in the standard (inverting) configuration. Then analyse your circuit.
You will see why in the first circuit you get gain=2 for positive halfwaves and 1 for the negative ones. (In the first circuit, output is at the cathode of D1.)
Regards
Z