I am unsure what a Bandonion does. Wheatstone concertina (yes, the guy who made the wheatstone bridge I believe) are also same note push-pull. More buttons. more. Wheatstone concertina sell for $10,000 or more. Melodions sell for $100
Zydeco typically uses the harmonica-aligned melodion. Bandonion music is what Astor Piazzola plays.
(can play a melodion. cannot play piano accordion or concertina)
Jimmy Shand is god. Just saying.
One confusion is that the word “melodeon” is used more in the UK, Ireland, and Australia (according to Wikipedia) and “diatonic accordion” in the US, but they’re words often used for the same thing, I think?
Harmonicas have free reeds and so do accordions and concertinas, but they don’t use the same parts as someone might guess from the way you said it. Harmonicas (and melodicas) have many reeds cut out of a single metal plate. (Or plastic for some harmonicas.) Accordions typically have pairs of reeds (one push and the other pull) held into a wooden reed block by wax, so they can be individually removed by melting the wax. Some less expensive concertinas use accordion reeds and the crazy-expensive ones use hand-made concertina reeds.
Also, a piano accordion or chromatic button accordion could be tuned to play some notes differently on push versus pull. (For example “Arabic tuning.”) This is rare, though.
A fundamental reason to have different notes on push versus pull is to use fewer reeds, saving weight and complexity, which is more important on smaller instruments. But this resulted in different music and playing styles too so they’re also culturally different.