The art of asking questions
I am always in awe of such in-depth questions... even more so when they are constructed so well. The art of asking questions is no less valuable than the art of answering them... and that is what makes CE EE so attractive.
I have developed the OP's valuable idea of comparing the three diode circuits by building, exploring and explaining them step by step. To show the evolution of the basic idea, I have inserted additional "shots" in this scenario.
I used to do it through real experiments (see my students story) and now with the help of CircuitLab simulations. To simplify the schematics and make them more conceptual, I have supplied the diode elements with a current source. In real circuits, its role is successfully performed by a humble resistor and voltage source because the voltage drops across the diodes vary little.
My explanations refer to the principles on which circuits built with discrete elements are based, not to technological considerations related to the production of ICs.
How to keep a constant voltage
This is one of the most important diode applications in electronic circuits where diodes stabilize the voltage by shunting; hence the name "shunt regulators". Let's follow the evolution of this powerful idea.
Constant resistor
The simplest way to create a constant voltage V = I.R (Ohm's law) is by passing a constant current I through a constant resistance R.
simulate this circuit – Schematic created using CircuitLab
The problem is when the current is "not so constant" or when a load is connected in parallel to the resistor.
Dynamic resistor
The remedy is very simple and intuitive - make the resistor "self-varying" (dynamic) so that its resistance changes in the opposite direction to the current and V = I.R = const. A very good way to understand this idea is to put ourselves in the place of the resistor and begin to perform its role. Let's do it together with the simulation below.
Imagine that I have reduced the current in half (5 mA) so the voltage across the resistor has dropped to 5 V. However, you sense this and increase the resistance twice (2 k) so the voltage remains 10 V.
simulate this circuit
Or if I have doubled the current (20mA), you halve the resistance (.5k) and the voltage remains 10V.
Diode
All types of diodes (ordinary Si and Ge, LED, Zener, Schottky, etc.) behave this way when forward biased. Let's for example investigate the IV curve of an Si diode in the schematic below by the help of the DC sweep simulation. The current through diode changes from 0 to 10 mA and for each step, the voltage across diode is plotted.
simulate this circuit
As you can see in the graph below, in the 0 ÷ 400 mV range, the diode behaves like an ordinary constant resistor with a very high static resistance (R = V/I). After that, however, it decides for some reason to reduce its resistance. As a result, its IV curve becomes almost horizontal and as they say, "its differential resistance tends to zero".
The problem is that the diode does have some series resistance R_S (for example, 0.568 ohm for 1N4148), and that changes the slope of the curve. Let's set it to vary as a parameter in the range 0 ÷ 100 ohms with a step of 25 ohm (significantly increased to see the effect).
Base-emitter junction
Interestingly, for better or worse, a transistor base-emitter junction behaves the same way as a diode....
simulate this circuit
... but it seems to have a higher series base resistance B_R (10 ohm for 2N3904).
Let's plot a family of base-emitter IV curves at different B_R as above.
"Active diode"
Basic idea. I don't know who came up with the brilliant idea to simply connect the transistor collector to its base (as later, the op-amp output to its inverting input)... but I am sure it was people familiar with Harold Black's idea of introducing negative feedback in amplifiers. This connection introduces a voltage-type negative feedback that has a drastic effect on the transistor behavior converting it from a current to a voltage "source".
Operation. At the first moment after turning on the current source, the transistor is off. The current flows through the base-emitter junction and the transistor begins to divert current through its collector-emitter part until equilibrium occurs (Ic = beta.Ib); the voltage settles at about 0.7 V. Thus the output collector-emitter part of the transistor is connected in parallel to the input base-emitter junction; so we can think of this combination as a "buffered diode". The true diode (base-emitter junction) diverts only a beta part of the whole input (collector) current; so it acts as a low power (signal) diode that determines the behavior of the power "diode". Most of the current passes through the collector-emitter junction that initially had the behavior of a current stabilizer but now acts as a voltage stabilizer.
simulate this circuit
Conceptual schematic. For the purposes of intuitive understanding, we can imagine the transistor as a "current divider" of two "resistors" in parallel - a low-power dynamic Rbe and a powerful controlled Rce, which interact. I had to play around a bit to adjust their resistances to match the real transistor schematic above. As you can see, Ic/Ib = Rbe/Rce = beta (140 here); the similarity is striking.
simulate this circuit
Now we come to the most interesting part of our study - to see what is the influence of R_B on the IV curve. As we expected, it is insignificant because the base current is "beta" times smaller than the collector current; the IV curves are almost horizontal and the same (as they say, the diode differential resistance in this part is lower). That is why the active diode is better than the ordinary diode. This confirms the assumption of @jp314.
Zener diode
Zener diodes have significantly higher voltage and power dissipation than ordinary diodes.
simulate this circuit
"Transistor Zener diode"
Therefore, it is of interest how to make a "transistor Zener diode". The recipe is simple: If we insert a Zener diode between the collector and the base, we will make the transistor raise its collector voltage with the Zener voltage (Vz is a disturbance for the transistor which it must overcome).
simulate this circuit
"Rubber Zener diode"
We can achieve almost the same effect with a voltage divider because if we apply not the whole collector-emitter voltage to the base-emitter junction but a part of it, VBE will be multiplied (like in the non-inverting amplifier). The "transistor diode" will act as a "transistor Zener diode" with any desired voltage. This network is widely used as a bias circuit in op-amp and power amplifiers.
simulate this circuit
The "rubber Zener" voltage can be smoothly adjusted by a potentiometer.
simulate this circuit
How to mirror the current
This the most important active diode application in electronic circuits (mainly in ICs) where a current from one place is "cloned" to another or more places. Furthermore, the direction of the current is reversed.
"Reversed transistor"
The negative feedback has the unique property to reverse the transistor behavior. Usually, the input voltage Vbe controls the output collector current Ic (see STEP 2.2) while here, thanks to the negative feedback, it seems as if the "input" collector current Iin controls the "output" base-emitter voltage Vout. This is because the transistor adjusts its Vbe voltage to pass the collector current supplied to it by the current source; so it acts as a log current-to-voltage converter.
simulate this circuit
"Normal transistor"
Then this "output" voltage can be applied as an input Vbe voltage to a "normal" transistor Q2 to convert it to another current Iout = Iin; so it acts as an antilog voltage-to-current converter.
simulate this circuit
Cascaded transistors
In this way, we obtained a "mirror" copy of the entering input current where the output current has the same magnitude but an opposite direction (again entering instead of exiting). It is interesting that the two cascaded nonlinear converters form a linear converter.
simulate this circuit
Note: I have used a little "trick" to adjust the collector current to exactly 10 mA - I have set a resistance of 860 ohm of the ammeter.