Be careful, your hurting your case, so by what you posted any antenna
can be a dipole if it meets the requirement of being fed two 180 out
of phase signals of equal amplitude?

no, there are several conditions, balanced feed is only one of them.


So, we all use improper dipoles, big deal. They are still dipoles,
spend some time building and using them, in real on the air tests, not
in your lab. Balun manufacturers won't make claims of significant
signal gains.

a balun isn't a gain device. neither is wire, but combine them, and you get
a gain device, which is a system called an antenna.

I'll keep building my dipoles without a balun, fed with coax, and be
very happy with how well they work.

ok