OK, here is a picture of what the tile cutter bit looks like. I am a bit ocd too so I keep every package for every bit and I store the bit back in that same packaging just so I will know which one is which.
This one is number 562, in the orange circle colored package.
Remember, cutting the door on the clay pot takes a while! It's not a quick job. You need to score the line first, then go back and gently start cutting with the side of the dremel bit. Don't try to force the bit through the pot, just press gently, back off a bit, press gently, etc. I usually alternate cutting one side then the other until I have just a small tab left at the center. My wrist generally gets sore after a while. Once you have just that small tab, don't try to just snap it off! You might end up with a disaster. Just gently work through it with the cutter until it falls off. I think I timed it one time and it took about 15 minutes to complete the door cut (that's without breaks). You will then need to go back and clean up the door way. I try to round all the edges off so there are no sharp points that the little critters might cut themselves on (or that I will cut myself on!).