What is the word used to describe the action of using the dental drill to ‘chip’ at the surface of your tooth?

You have clearly done some research or thought on the matter but have missed the obvious abrade

Abrade = to remove part of the surface of something by rubbing

Cambridge dictionary

The abrading smooths the surface. If the abrasion is done by the rubbing of a very finely textured drill surface it will result in polishing of the tooth.

'Chip', 'saw' and 'drill' are all terms drawn from craft and industrial processes working on wood, stone and metal and all predate the invention of the dentist's drill. It is, however, appropriate to use craft and industrial terms to describe some of the processes in dentistry although the terms can be quite disconcerting when it's you teeth that are going to be subjected to the process.

If a piece of mental, say, has a ragged edge (perhaps from a saw cut) then a metalworker will often smooth it down either with a file or a grinding wheel until the edge is smooth. There are a number of terms for this process:

File:- This is usually a manual process using a hand-held file so is not appropriate for the dental equivalent.

Sand:- The process of using an abrasive sheet (originally sandpaper) to remove cutting marks and small irregularities from the piece. Would not be used in dentistry as there is no abrasive sheet used.

Grind:- This is done with a rotating stone driven in the past by a treadle but now usually by an electric motor. This is an accurate description of what is done with the drill but is quite a scary thought when it's your teeth.

Polish:- The process of smoothing down a worked piece until the shaping marks have vanished and, often, the surface is smooth enough to reflect light. Sometimes polishing is carried out manually, sometimes with a machine. Polishing in dentistry, however is usually applied to a finishing process used after plaque has been removed from the teeth in a 'scale and polish' procedure.

Buff:- This is a similar process to grinding and to polishing in that it is done using a rotating wheel but is much finer and is usually applied to the final stage where a really smooth finish is required. Quite often a buffing wheel will be covered in soft fabric, possibly even a fleece of some kind.

There are other terms for this process which have been used in industrial environments. Examples are:

Dress:- This can cover grinding, polishing and buffing as a series of processes. The idea of dressing a piece is to smooth its surface and make it presentable

Fettle: Mainly a British English term in this sense, and predominently northern English dialect at that meaning 'make ready'. In many situations (particularly metalworking) it can be considered a synonym for 'dress'

If the dentist is taking off a rough edge from the tooth and, perhaps, shaping it a little the most accurate description is probably grinding. The primary objection to this use is that "grinding one's teeth" describes the process of someone moving their jaws so that the teeth grind against one another. However it is probably the best word for the process. An older person from the old industrial north of England might say that he had had his teeth 'fettled' but I don't believe any dentist would normally use the term.