I have the impression that the circuit depicted with the photodiode in the app notes is for high frequency applications, like fibre optics and such. You stated in your post you need this quick responsiveness, but how quick is quick?
If you're talking about milliseconds I wouldn't go with this type of circuitry and a photodiode. The people who wrote the app notes are experts and those transimpendance amplifiers look quite nifty. The components need to be picked carefully (low-noise, FET inputs opamps), proper shielding needs to be foreseen, PCB layout,... All come into play to get a stable circuit.
If you have a look in the same document at the circuits depicted with a phototransistor, they are much simpler. Basically it's a voltage divider you directly connect to the ADC. You can put an amplifier in between, but it's just a regular non-inverting opamp.
In regard to the difference in signal amplification. This is taken from wikipedia, it's the current characteristic that is linear and not the voltage.
In electronics, a transimpedance amplifier, (TIA) is a current to voltage converter, most often implemented using an operational amplifier. The TIA can be used to amplify the current output of Geiger–Müller tubes, photo multiplier tubes, accelerometers, photo detectors and other types of sensors. Current to voltage converters are used with sensors that have a current response that is more linear