Time for a history lesson!
The first transistors that would be recognised as bipolar junction transistors are the grown-junction transistors.¹ These were made by growing a germanium (later silicon) crystal while changing the composition of the melt it's grown out of by adding dopants. This works, but the transistors produced have poor characteristics by modern standards and they're quite expensive.
Later, alloy-junction transistors would be made. These were produced by taking a piece of doped semiconductor and fusing beads of dopant metals of the opposite type onto it, where the alloyed areas would act as the opposite-type semiconductor to form a PNP or NPN structure. This allowed much better performance than the grown-junction transistors, but it was still an inherently slow process as each transistor had to be made individually.
Next came diffused-base transistors. They started with a substrate of appropriately doped semiconductor and then exposed it to a hot gaseous dopant material in an otherwise evacuated chamber, creating a region of opposite type on the surface. One of the two output terminals, either the emitter or collector, was still applied like an alloy-junction transistor, but the other was not.
The double-diffused transistor was produced shortly after, when it was realized that the diffusion process could be done with two dopants at once of opposite types, relying on the different diffusion rates of dopants of different atomic mass to get things to line up properly internally. This produced very good transistors for the time, and similar processes are still in use for certain things today.
The early double-diffused transistors were known as mesa transistors, because the transistor itself was formed on a raised part of the substrate² (Or possibly parts of the substrate were etched away after the transistor was formed; I'm having trouble finding clear sources). This caused some reliability problems, which were largely solved by the invention of the planar transistor in 1959.
The planar transistor was a major breakthrough, though not particularly due to its planar shape. In previous transistors, the oxide that formed during processing (due to heat and exposure to oxygen) was removed out of a concern that it could contaminate the transistor. This actually proved counterproductive (at least in silicon transistors, which were soon to take over from germanium entirely); the planar process, in which the oxide is just left there, proved to be much more reliable as the oxide actually protects the junction from outside contaminants. Planar transistors were the first ones that didn't need to be hermetically sealed in metal can packages (though it would be a while before plastic packages became the norm).
Planar transistors are still the type of BJT used most often today, but there's one more step before we get to the devices you're asking about:
Separate from the other developments here, a process called epitaxy was invented in the early 50s and perfected (or at least brought to a commercially viable state) in 1960. Rather than starting from a uniform semiconductor, you can grow a very thin layer of differently-doped semiconductor crystal on top of your existing semiconductor wafer. This may sound similar to the grown-junction transistor mentioned earlier, but that used a process that couldn't be controlled very precisely--epitaxy involves growing the crystal through chemical vapor deposition, and produces more uniform and thinner films than the grown-junction process ever could. This was combined with the planar process to produce epitaxial planar transistors, which had the much better reliability of planar transistors combined with excellent performance (both improved switching speed and better blocking voltage) from the epitaxial process.
Epitaxial planar BJTs are still common today, some sixty years after their invention. Most discrete transistors you buy today will be either epitaxial planar BJTs or a completely different type of transistor called a VDMOS, a type of MOSFET. The processes for making integrated circuits also developed out of the epitaxial planar process (though they changed quite a lot over time).
Notes:
¹ The first transistors ever made were point-contact transistors, but they're somewhat different from BJTs in ways I personally don't fully understand, so I'm not going into any detail here.
² I'm not quite sure why it was like this; none of the sources I can find say. It may have been a way to isolate adjacent transistors from each other so that they could make many on the same wafer, keeping any sawing to the areas between transistors instead of trying to cut one transistor into two (which could cause problems on the cut edges). That is only an educated guess, however.
Sources used: Primarily the Computer History Museum's timeline of semiconductor breakthroughs and the various articles linked in the History section of the Wikipedia page on BJTs and their citations.
