How do I determine what kind of replacement shifters I need?
You are correct that 2x8 refers to 8 speed cassette or freewheel and a crank with 2 speeds. The one that can be confusing is 18speed which can be an older 3x6 or a more modern 2x9. The other concerns is mountain or road drivetrain and Shimano, Sram or Campagnolo  compatible. You can search this site for compatibility questions that may match your specific circumstances.