"... When carbon burns in oxygen, that reaction produces carbon dioxide and water.."
'Water'? That would be interesting. With just carbon and oxygen present, where did the hydrogen necessary for water come about?
Technically, with a diamond, wouldn't the very terminal carbons of the carbon-matrix which are in contact with atmosphere contain a hydrogen atom, as the carbon-carbon tetrahedral structure must end at that point? Either that, or perhaps a hydroxyl group. Hydrogen sounds most likely. Certainly would only produce an ultra-trace of water, but an ultra-trace is not nothing!
More importantly, as I think of "ideal" diamonds being a crystal of pure carbon, each atom in sp3 orbital links to four others, if it is pure and without cracks, aren't you looking at a single molecule of carbon, since all of it is covalently bonded? This is the ideal of course. Inclusions muck up the notion, but not if they are trapped and there are no surface-to-surface cracks in the diamond, it seems like each could be a single "molecule" of carbon.
Diamond is s stable to 1000 °C in air, 1500 °C in inert or reducing atmosphere. No glow to ~1000 °C re Debye temperature. If you drop a really hot diamond into liquid oxygen, it skittles around burning.
However, in contact with carbiding metals, diamond catalyticaily decomposes to graphite at room temperature, If you put some diamond dust on your finger and draw ia across molybdenum, you can see the immediate darker grey streak from molybdenum carbide.
A poorly cut diamond will  cleave into thin triangles with hardly more than a brisk tap.
Yes, diamond can be burned. The most common form of burning in everyday life is carbon combustion. In carbon combustion, carbon atoms break their bonds with each other and with other atoms (which requires energy) to form bonds with oxygen atoms (which releases even more energy than first required). The net extra energy released in this reaction can then go on to rip more carbon atoms free of their non-oxygen neighbors, thus continuing the reaction, or the energy can escape as heat and light. Carbon bonded with oxygen makes carbon dioxide and carbon monoxide, which are the waste products of combustion. They drift away from the fire as gases. Since biological organisms, and the fuels derived from biological organisms, contain a lot of carbon, much of the burning that goes on around us is carbon combustion. This includes camp fires, candles, car engines, and gas stoves. Pure diamond consists only of carbon atoms bonded into a dense, strong crystal lattice, so diamond can also undergo carbon combustion. In fact, Antoine Lavoisier first determined that diamond is made out of carbon by burning it and showing that the combustion product was carbon dioxide. The book Diamond Films and Coatings by Robert Foster Davis states, "In 1772 the French chemist Antoine L. Lavoisier found that the products of diamond combustion behaved as, and most assuredly were, solely carbon dioxide... He determined that the ignition and burning of each produced equal amounts of ‘bound air'."