Does a laser “etch” things, or does it “engrave” them?

I think which one you would use also depends strongly on the context in which you are working.

If you are working on marketing material, for instance, I'd probably lean towards "laser-engraved." On the other hand, if you are writing a technical work, then I would comment that a search in Web of Science (note: subscription required), actually returns more than an order of magnitude more links for "laser etch*" (nearly 40,000) versus "laser engrav*" (about 3,000). So, for any sort of technical writing, laser-etched is actually preferred to laser-engraved.

A Google Scholar search is even more lopsided in favor of "laser etching":

• laser etching: 639,000 results
• laser engraving: 17,000 results

Engraving is a process of creating depth patterns in a surface by direct incisions. Etching is a form of surface marking done via chemical corrosion, which if prolonged, can definitely result in deep markings, but otherwise etching usually results in a textured effect (googled etched glass). Lasers are useful in that there is great control over what you want to accomplish be it an textured effect similar to chemical etching or deep engraving and even cutting all the way through the material.