I grew up in "olden times"? I'm technically (barely) a millennial! :(

The latter being short for "gyroscope" of course.

It can also be a 'copter with an unpowered main rotor, a gyrocopter/autogyro/gyroplane or "gyro" for short.

Do you mean a "'pter"?

Helicopter is (like gyro, from the Greek) a helico-pter (whirl wing)

You have the right derivation and you've broken "helicopter" correctly, but what exactly is a "gyroco"? It seems to me "gyrocopter" was very likely coined in English from the root "gyro" and the "copter" from "helicopter" as a back-formation.