if I were to convert back to having a starter button (because hey, it's cool!) are the buttons different, like you said, or universal, like Gordie says above?
They are definitely different. It depends on if your starter is Autolite or Delco-Remy.
The Autolite type, KF P/N 212059 has one wire connected to it and simply grounds the solenoid. The button is this one -
http://ecatalog.smpcorp.com/bwd/#/vehicles/parts/num/PB2?type=sAutolite starter (white wire goes to starter button):
The Delco-Remy style, KF P/N 201464, has two wires connected to it and supplies voltage to solenoid.
It's also a much heavier duty push button switch as it has to pass more current.
Delco-Remy starter (white wire goes to starter button):
Gordie is correct in that you could use the Delco two wire style in the Autolite equipped cars. You just need to connect one wire to the solenoid and the other wire to ground. My '52 Henry J Vagabond was that way for a long time till I found these Ford units were the same and got one. You cannot however, use the one wire Autolite style in a Delco equipped car.