Biggest problem is getting started, as in tile, You can use a diamond point scribe to break the vitrified surface coating in the center of the hole, then drill with a clay dam around the area to hold the water or kerosene or your favorite coolant. Make sure that you have a large enough pool of coolant. You should also drill only half way and do the same from the other side to prevent break through.
Using a smaller sized drill bit then finishing the hole to size can help. I have some small sized diamond coated hole saws. That work well in the vitrified materials. I also have found that a slower light pressure works better than fast and hard.
Don't know what version of a "tile drill" you are using,
I have had success with this type.
You can also start the holes with this type of burr too
I also have a small dia. set off carbide/diamond hole/core drills/saw that work with flood cooling. Again working from both sides doers help on some items.
Using a slightly large baking/cake pan to contain the cooling fluid works well and keeps the drill press cleaner.
Don't have a photo of the core drills.
I'm not pushing the HF tools as a an item for occasional use they work fine. But they have good photo's of the type.