Which particles are classified as PM2.5? How exactly is this defined?
I think answering your questions in reverse will make more sense. The "size" of PM is typically the aerodynamic/inertial impaction size, as you guessed. My standard reference for this is this paper. Their figure 8 is a rough schematic of where particles of different size deposit in the lungs. Things with very large or very small inertial impaction size get filtered out by the nose and upper respiratory system. (The "very small" inertial impaction I would consider as the fuzzy boundary between particles and gases, and gases adsorb onto mucous.) It's the particles with an aerodynamic size smaller than 2.5 microns that follow the airflow all the way from the nose and mouth into the lungs. So from a health perspective, aerodynamic size is what matters most.

Similarly, most measurements of particulate matter filter them by size first, and those filters are based on how much they deflect when the airflow has to curve around a bend. So when measurements focus on PM2.5, that's also anything with aerodynamic radius less than 2.5 microns. They might later do optical scattering or other analysis, but AFAIK the aerodynamic filtering always comes first.

I would say that PM2.5 is everything 2.5 microns and smaller, and PM10 includes all of PM2.5. When scientists want to differentiate the two, we usually use the terms for the different "modes": coarse mode for ~1-10 micron, accumulation mode for ~.05-1 micron, and nucleation or Aitken mode for anything smaller. (Fine particulates are nucleation+accumulation.) The reason for PM2.5 including everything is that PM2.5 standards are focused on the mass concentrations, usually in micrograms/cubic meter. The total mass in the nucleation mode is typically negligible compared to the other two. (The number concentrations particles in the nucleation and accumulation modes might be similar, but the masses are way different.)

Edit: I don't think I made this clear, but the reported air quality value for PM10 would include PM2.5, but if something is reported as "coarse particulates" it would be (PM10-PM2.5). You may also see a quantity TSP - total suspended particulates - which may include things even larger than PM10 that are being kicked up by particularly strong winds, or smoke and ash from nearby wildfires.