Been there, done that, but frankly the EE courses I took in college were a lot more detailed.
You're ignoring the base of the transistor.
How so?
I think you're misreading the point being made. A transistor is "back-to-back diodes" in the sense of its internal construction at the semi-conductor level. A diode is a single PN junction:
That is, a section of P-doped semiconductor connected in series to a section of N-doped semiconductor.
A transistor is an NPN or PNP sandwich:
If you take two NP junctions (a diode) and join them head-to-head or tail-to-tail, you get a sequence of either NPN or PNP -- the structure a transistor. The middle of the semi-conductor "sandwich" is the Base of the transistor.
Note that this is very different from AndrewC's attempts to divert the discussion to the straw man version of "soldering the leads of two diodes together", which of course is quite another thing.