No, not drill bits. Keep a pool of water such that the
boring bit is always wet and it will last for many holes. It doesn't mean a lot of water, just more water than will go dry and let the bit heat up. No heat = long life. Often, there is a concave bottom of the pot that will hold a couple ounces of water which is enough. And guide, don't force. Let the bit rest in place and "wear" its way through. Don't apply any more downward force than it takes to keep the bit in-place. Starting is the hardest part. Once a groove is there the bit will follow it down. If you can trap the pot in place so that it doesn't move when the bit touches it, that makes things go better. If it takes you an hour to jury-rig a fixture that traps the perimeter of the pot between boards clamped to a flat surface so you can concentrate on keeping the drill vertical and starting at low RPM to make the initial groove, that's a good use of your time. Drilling trough will only take a minute or so, but a clean cut is the product of setting up the job so that it will go smoothly.
Boring bits are hole cutters with diamond or carbide grit on the outer diameter and are much easier to use than drills made for concrete. And cheaper, too. They grind a path through and do not break through the far side the way a drill bit does.