Not finding fault with your explanation, just adding my own color:
The subject is "active/powered speakers", but we are talking about powered speakers. Further, there are no "active speakers", there are active (crossover) filters. That is, there are passive filters (made of capacitors, inductors, and resistors), and active filters (which contain gain elements, resistors and capacitors).
Active filters are more flexible in a number of ways, and don't lose signal level—and eliminate inductors entirely, the most annoying component—but it's impractical to feed them with high voltage signal levels and power. Passive filters only need the proper ratings on the components, and don't need to be powered. So, if a power amp is feeding a crossover (in a passive speaker system), it's going to be a passive crossover. And if the speaker is powered and has drivers over multiple bands, it's almost certainly an active crossover feeding multiple amps.
Of course it's possible to have a powered multi-way speaker that has a single amp and a passive crossover, but usually powered multi-way speakers want to have efficient amplification (for heat purposes, at minimum).
And, of course, it's possible to have passive speaker enclosures fed by multiple amps, fed by active crossovers. Probably still used in large-scale professional sound (concerts), and at one time in studios. I used to use (still have) 8-inch 2-way speakers with separate 12-inch woofer cabinets, using a rack-mount active crossover to amp the 2-way cabinet and the woofer cabinets separately—so there are both active and passive crossovers involved.
But really we're either talking about power speakers (that happen to have active crossovers), and unpowered speakers (that happen to have passive crossovers). So "active" is pretty much a moot point for the purposes of this thread.