There are different types of nine-button accordions. What most people see these days is one that is shorter than a ten-button accordion and a little wider. The other type is mostly associated with accordion builder, John Hebert. Mr. Hebert built a regular sized accordion, but just left off the top button. One of the first accordions I learned to play on was an "Hebert." There aren't many around, but I think that Jamie Berzas still plays one regularly.
As for volume and such, I think the same factors that create noticeable differences in accordions from the same builder also stand true here. I'm sure there are some scientific equations to calculate the difference in air volume and pressure created by the different bellows sizes, but personally, I have not noticed any big difference.