Where did Bootstrapping get its name?
The term is derived from "pulling yourself up by your boot straps".  In engineering it usually means to start with little, use that to make a little more, use that to make even more, etc, until eventually to get to full operation.

From that, it is sometimes used to refer to any startup process.  When a computer is first turned on, it goes thru a "boot" or "bootstrap" process to eventually load the operating system and be ready for use.  Now, "booting" a computer is synonymous with starting it up.

The computer actually goes thru several startup phases before being ready for normal operation, so this is a valid use of the term.  I remember computers from the 1970s where you had to first toggle in a small program of maybe a dozen instructions.  That was just enough logic to read a more complicated loader from a paper tape or something.  That load contained the more substantial routines to read from the disk, etc.  Modern computers still go thru a number of startup phases.  The initial one is read from a ROM, which is why you can simply turn it on and wait.  Booting a computer has actually gotten more complicated and takes longer than it took the much slower and smaller computers of the 1970s to boot up.

Personally, I wouldn't call a emitter follower a bootstrap circuit.  I think that's a misuse of the term.

A valid example of that term in electronics is referring to the circuit that provides the initial internal power of a line voltage to DC power supply.  Ultimately the supply will be a switching regulator that transfers significant power from the AC input to the DC output.  However, that switcher needs to be powered somehow.  In other words, you need a power supply to power the power supply.

The bootstrap circuit in a power supply provides the initial slug of energy for the switcher to run until it is able to make low voltage DC efficiently and at significant power levels.  This includes providing the switcher's own operating power.  The bootstrap supply is usually quite inefficient, but simple enough to run without itself needing a power supply.  Either that bootstrap supply uses very little power, or it is shut off once the main power supply is up and running.