Sorry, those circuits are a bit of a jumble as they have been used for several explanations - single rail, dual rail high speed, low speed etc.
Normally I would reverse bias the photodiode to reduce capacitance although for large area devices the leakage under reverse bias can be more of an issue. The bias voltage to the non-inverting input can be 0V if you use dual rail. The non-inverting input and photodiode don't have to be the same.
If ambient light could saturate it then you need to consider modulating the light source, keeping the initial stage gain low, AC coupling and bandpass filtering.
Keith