What are the differences between optoisolator output styles?
Phototransistor
This is the most basic variant. When there is an input signal, the phototransistor switches on like a normal transistor, i.e., creates a low-impedance connection between its collector and emitter (up to a certain current limit).

However, a transistor optocoupler does not amplify signals as much as a normal transistor. Typically, the ratio of output current to the LED input current (CTR = current transfer ratio) is about 100 %, i.e., there is no amplification at all.

Phototransistors have a very large collector-base junction (to be able to catch much light), which implies a large collector-base capacitance, which makes phototransistor optocouplers comparatively slow, especially when switching off from saturation.

Phototransistor optocouplers are cheapest, so they are used unless some other type is needed.
Phototransistor with base
On optocouplers with a base pin, it is possible to connect the base to the emitter through a large resistor (typically 1 MΩ). This allows the charges in the base to be removed faster when the transistor needs to be switched off, i.e., switching off happens somewhat faster. (Also, switching on is delayed by a little bit.)

It would be possible to inject feedback into the base pin to speed up switching, but this is hard to do in practice because of large manufacturing variations that result in very loose CTR specifications.

When the base pin is not used, it might pick up noise (depending on the environment).
Darlington
This is essentially a phototransistor with lots of extra amplification.
Typical darlington optocouplers have a minimum CTR of several hundreds percent.

Darlington optocouplers work with very small input currents, but they also amplify noise, and having two saturated transistors makes the time needed to switch off even larger than with a single transistor.
Darlington with base
See phototransistor with base.
Photovoltaic
Photovoltaic optocouplers do not switch a current between their output pins, but just use many photodiodes to generate a current. There is no transistor for amplification, so this current is very small.

Photovoltaic optocouplers are typically used to charge the gate of a FET.
Photo FET
This is a photovoltaic optocoupler with built-in FETs.
Two FETs make it possible to switch AC current between the output pins.
Phototriac/SCR
Allows to directly switch an AC current.
Typically allows less current than a photo FET, but is cheaper.

(A common way to switch a large AC load is to use a small phototriac to switch a large triac.)
linearized optocouplers
Optocouplers have large CTR variations due to manufacturing deviations.

Linearized optocouplers do not have much tighter specifications, but they have two similar photodiodes that generate two similar output currents. One of them can be used to construct a feedback circuit to control the input signal to get the desired linear behaviour.

However, in practice, the most widely used mechanism to transfer an analog signal is not through a linear optocoupler but with a PWM signal.
high-speed/digital optocouplers
The linear behaviour of phototransistors is often not needed.
Digital optocouplers use more integrated components (e.g., separate photodiodes, non-linear amplifiers, and/or Schmitt triggers) to allow faster switching.