In the modern aviation context, is the word "supercharger" generally assumed to mean a gear-driven system rather than an exhaust-driven system?
"Turbosupercharger" was General Electric's official term for their system, probably the first mass produced turbocharging system and was used on a range of US fighters and bombers (the system was very sensitive to back pressure in the turbine outlet which could promote stalling, and so they either had the turbine exposed like on the P-38, or if there was an exhaust duct it was very very short).

Anything that compresses the intake charge is a supercharger. Prior to the GE system, they were all mechanically driven and the mechanical drive naturally had horsepower losses associated with that type of drive.  GE added the word turbo to differentiate between the common gear driven or direct driven supercharger and their new and novel system that eliminated most of the energy losses from compressing air.

Move forward into the post war era and the adoption of systems to cars and light aircraft and people get annoyed having to say this pain in the butt 6 syllable word and start to cut out the "super" part, because it doesn't really detract from the meaning, and there you are with "turbocharger" for both cars and airplanes.

Meanwhile, "supercharger" stays as it always was and remains associated with mechanical systems, because the terms, although incomplete technically, don't contradict each other on common usage they continue.