The Wii Nunchuk has exactly two buttons—not three, not one. Here's the breakdown:
There is no third button. The joystick is analog input, not a button. The Nunchuk also contains an accelerometer (motion sensing) but no rumble motor—that's in the Wii Remote.
Common confusion comes from the Wii Remote having many buttons (A, B, 1, 2, +, -, Home, D-pad, Power), but the Nunchuk is deliberately minimal.