This page calls it a transistor based voltage divider. Is that the same thing?
I don't know because the page would not let me see anything. I have never come across it being called that but it could make sense.
If it is current controlled then how do you know when it is saturated?
It is saturated when an increase in base current produces no increase in collector current.
For instance if I had a load on transistor that took 100mA, but my base current and gain were such that only 50mA were allowed to flow I am assuming that would not be saturated.
But does 100mA constitute saturated, or 150 or 200mA? Like at what point is it saturated?
When a transistor is saturated there is a fixed voltage between emitter & collector, this is normally 0.7V although at high currents this can increase, in the data sheet this is called Vsat. We say the transistor is "Hard on" in other words the rheostat is turned down as much as it will go.
So the saturation current for any circuit is the voltage across the load resistor over (divided by ) the resistance of the load resistor. The voltage across the load resistor is the supply voltage minus the Vsat voltage.
Like in an op amp or something?
In fact transistors were around first and op amp feedback used to be described as "like the emitter resistor".
So the less the base current the more the resistance?
Does the higher impedance (then lower) impedance have an advantage?
The bigger a capacitor is the more expensive it is (in general ) so it is better to have a high resistor and a small capacitor. However such a high impedance will not drive very much, so in a simple RC filter you have to have the impedance low enough to drive the load. The transistor converts the impedance so you can have a low impedance output to drive a load.