Learn English – Etymology of the word *plane* as used in *airplane/aeroplane*

I was curious about this etymology since plane retains a distinct meaning that only applies to airplanes whereas other words like aircraft, airship, airliner, air-vessel, air-boat, aeromotive, etc. all have meanings of the non-air (or aero) part of these words/phrases that are more generic.

Wikipedia is uncertain if the word is derived from Latin or Greek:

comes from the Greek ἀήρ (aēr), "air" and either Latin planus,
"level", or Greek πλάνος (planos), "wandering".

Wiki's sources for Latin are Merriam-Webster Dictionary, and for Greek are both Oxford Dictionaries and A Greek-English Lexicon.

Dictionary.com cites the 2017 Random House's entry with the opinion that plane may be from the sculptural aspect of the aircraft as a relative flat thing similar to forme plane:

1870-75; < French aéroplane, equivalent to aéro- aero- + -plane,
apparently feminine of plan flat, level (< Latin plānus; cf. plain),
perhaps by association with forme plane; apparently coined and first
used by French sculptor and inventor Joseph Pline in 1855

The 2017 Online Etymology Dictionary by Douglas-Harper argues that there was a pre-existing meaning of any heavier-than-air flat thing (like Beetles' wings) to be called planes:

1866, originally in reference to surfaces such as shell casings of
beetle wings, from French aéroplane (1855), from Greek-derived aero-
"air" (see air (n.1)) + stem of French planer "to soar," from Latin
planus "level, flat" (from PIE root *pele- "flat; to spread"). Later
extended to the wing of a flying machine. Meaning "heavier than air
flying machine" is first attested 1873 and probably is an independent
coinage in English.

But the entry then adds the comparison to the Greek almost as a way of sort of hedging it's bet:

Greek aeroplanos meant "wandering in the air," from planos "wandering"
(see planet).

I also found this explanation attributed the American-Heritage Dictionary:

plane in the sense of “winged vehicle,” first recorded in April 1908,
is a shortened form of aeroplane. In June of that year plane appeared
in a quotation from the London Times that mentioned one of the Wright
brothers. Aeroplane, first recorded in 1866, is made up of the prefix
aero-, “air, aviation,” and the word plane, referring to the structure
designed to keep an air vehicle aloft. Originally the plane in such
contexts was imagined as flat, hence the choice of the word plane; in
practice this surface must curve slightly in order to work. The word
aeroplane for the vehicle is first found in 1873. The first recorded
appearance of the form airplane in our current sense, which uses air-
instead of aero-, is found in 1907. An American flies in an airplane
while a Briton still travels in an aeroplane, but both can catch a
plane.

From all of these, I see three distinct theories evolving for the word plane as it is used in airplane/aeroplane:

1. Plane, like planet, is derived primarily from the the Greek word planos meaning wandering (per Wikipedia quoting Oxford Dictionaries and A Greek-English Lexicon; also alluded to by Online Etymology Dictionary)
2. Plane, like plain is derived primarily from the Latin plānus and the primary feature is the sculptural features of the wings being flat and level as in forme plane (per Wikipedia quoting Merriam-Webster Dictionary; per Random House Dictionary; per the attribution to American-Heritage Dictionary)
3. Plane is derived primarily from the French planer meaning to soar (per primary theory given by Online Etymology Dictionary)

Anyone able to clear this mess up?