Darlingtons, MOSFETS, and bipolar junctions
Your question distills to "how should I use transistors?" and the answer
fills whole books. Distributors carry tens of thousands of different transistors to meet the needs of designers. As a consequence, I don't believe you can create an infographic that is complete, correct, and useful at the same time.

What I can give you is a set of heuristics:

Choose...


...a BJT when:


you need a current amplifier¹
you have power to burn, so the required base current doesn't bother you
the "bottom leg" of the transistor can get pulled above the "top leg," and you need this path to not conduct when that happens, as it would in a MOSFET due to the body diode
you need to exploit the C-B or E-B diode²

...a JFET when:


you only need small-signal amplification³
the base current of a BJT would be a problem in your application⁴
you need a voltage-controlled resistance
you need higher transconductance or lower flicker noise than you'd get with a MOSFET
you need a small, discrete, low-component-count constant current source

...a MOSFET when:


you need a switch⁵
you need higher input impedance than a BJT, and can't find an IGBT that fits the bill
you can't find a JFET studly enough to control the power your circuit needs⁶
you were looking at JFETs, but then realized you needed enhancement mode operation⁷
you want to exploit the inherent body diode as a feature, rather than consider it a problem to cope with⁸

...an IGBT when:


you need something that behaves like a high input impedance BJT



I've largely ignored transistor combinations above, because once you start combining transistors, the possibilities literally become infinite. Today you have a range from Darlington pairs to billion-transistor ICs, with new circuit arrangements being designed all the time.

There are many, many more types of transistors, but for the most part, you want to figure out all the above before moving on to them. The others are usually derivatives of the above classes, so without the initial grounding, you will have no basis to select one of the exotics. Examples:


Need to exploit the fact that a BJT's base is light-sensitive when not encapsulated? Okay, that's a phototransistor.
Need multiple emitters or collectors on your BJT? Sure, no problem, that’s easy to create in silicon; it’s often useful and therefore often seen at the custom silicon level.
Need a BJT, but a whole lot faster? Okay, we just won't use doped silicon for everything, we'll combine materials into a heterojunction transistor.
Need to pack transistors in even tighter than normal, but running into trouble with the short channel effect? Okay, we'll give you a FinFET.
Need something that works kinda like a diode, and kinda like a BJT? Okay, we'll blur the lines for you and give you a UJT.
Need something like an IGBT but with lower loss and higher current gain? A SiC junction transistor might be what you're looking for.




Footnotes


You can also make voltage amplifiers with bipolars, of course.
Small-signal BJTs make great low-leakage diodes.
JFET-like power devices exist, such as  Infineon's CoolSIC, but they're relatively new and exotic, thus expensive.
This is why there are so many JFET-input op-amps.
All the other transistor types can act as a switch, too, but MOSFETs excel at this.
A power MOSFET isn't quite the same thing as a power JFET,³ and it isn't the only alternative, but it is often the simplest, cheapest alternative.
JFETs are inherently depletion-mode devices, only.
One my answers here at EE.SE shows such a circuit.