That's not quite right, but close. The quartz itself does not have a connection and the two connections are not an anode and a cathode (which inplies a polarity) but electrodes bonded to the quartz to induce movement and pick up a voltage as the crystal relaxes. The resonance is due to the mechanical properties and size of the quartz, applying a voltage across the electrodes causes a physical change in the crystal lattice and on removal of the voltage the lattice reverts to it's original position, releasing a small volatge as it does so. The critical thing is that it will only do this at a particular frequency. You can reverse the connections to a quartz crystal and it will work exactly the same.
Three pin devices are usually resonators, they work in the same way but use ceramic materials instead of quartz and are not as temperature stable. They also have lower 'Q' meaning they can be persuaded to resonate over a wider range around their design frequency. They also have two pins but as in many case they are used in microcontroller applications where the frequency may not need to be very accurate but the cost must be as low as possible. As most microcontrollers have built in oscillators that rely on an external resonantor or crystal and also need loading capacitors from each end of them, some manufacturers build the capacitors inside the resonator body, bringing their common connection (normally connected to ground) to a third pin. It's a cost saving convenience to designers.
I've attached a picture of a glass cased quartz crystal that will help to explain. Sorry it isn't very clear but it's a small crystal and taken on a mobile phone! You can just see the circular disc of quartz, almost touching the glass walls, the lighter parts are the electrodes, the back one is visible through the quartz. you can also see the tiny solder jonts where the wires are bonded to the electrodes.
Brian.
