It certainly makes one think and get back to basics that many of us have either never studied or brushed aside. Most of the text books you refer to properly asssume basic knowledge.
The discusion here is off track in the fact its confusing self inductance where the current flowing in a coil by itself can and will produce a flux. (note flux is like current mmf is like emf )this flux cuts coil creating a back emf - opposition
Also Mutual inductance is mentioned this is where th above flux cuts another coil -transformer
Neither of the above are in a generator as the flux flowing can only come from the rotor, if mmf from any coil equals rotor mmf no flux no induced emf.
In your twin coil a mmf is set up by each coil but must both be in a common direction to oppose the one rotor mmf and resulting flux is reduced. Its not relevant that the coilsare wound in different directions currents will flow to create opposing mmf to rotor and as they can not produce a actual flux themselves self inductance is not a issue. bifolar is a means canceling self inductance . Here we have inductance yes but due to rotor flux. Its much easier to follow the theory in transformers where the flux is produced by the primary winding. Read up sometheory on transformers, I just looking though my old books most authors do avoid discussing fundementals of inductance in generators.
another term i have seen here armature reaction was originally given to DC machines its the distortion of the rotor flux due to the opposing mmf it caused the brush position to be incorrect as load increased. only effect in our generators is crowds flux to a smaller pole face in fact with air cores no real effect.
Leakage flux is when the opposing mmf makes it easier for the rotor flux to bypass the coil. All the effects tend to reduce output voltage as load comes on. Inductance of the rotor coil combination is the main cause though in our small units, we often kill the rotor flux creating the flatterning of of out put current.

Enough .......