Photodiode in photoconductive vs photovolatic configuration
Right, except for photovoltaic it's not leakage current... just photocurrent.  The biased version will give a little more dark current.  The big advantage of biasing the photodiode is that you reduce it's capacitance.  (typical numbers might be by a factor of 4 or 5. at 10V) And this makes the detector faster.  Which is almost always a good thing.  Oh one final disadvantage to the bias case is that any noise on the bias voltage appears on the output.  (so filter the bias supply.)