One issue, however, is by including 240/120V as two phase system in the family of poly phase systems, we deprive some of the advantages of having poly phase systems such as lower cost, longer life, lesser vibration etc., which rightfully belong to poly phase systems such as 'true' two phase motors and three phase motors.
A system has its characteristics. We observe and make use of them as we can. We discover and utilize what already exists in the physical blueprint of nature.
Balanced two phase is different from quadrature two phase. It exists as part of a balanced polyphase system not by our design but because it already is by nature.
We can say different systems have different characteristics but a balanced two phase system is part of an ordered system whether we like it or not.
Quadrature two phase is not a balanced system but we don't kick it out of the polyphase group. It gets an asterick because it is really a subset of a four phase system which is a "true" balanced polyphase system.
Again, definitions matter. For a polyphase system using the canonical definition of phase system orders, balanced two phase is included and quadrature two phase is not. For systems that have the advantages you discussed it is the reverse.
However, balanced two phase has advantages that quadrature two phase does not. That does not change what they are but shows how we can use them.
You can define things however you want but it does not change nature. So pick the definition you like and be happy with it. Or be satisfied that each definition is valid in its own right. I promise the world will continue to spin its life cycle regardless of the choice you make so no worries.