You need to go to the @Joseph Corlett, LLC school of sinks... he has many many posts you can search on here about these issues. For installation he favors a positive reveal as in your first photo because the seam does not get gunky.

The first photo is an undermount and second photo is a top mount, which is something a different animal. So if your question is more about those types (as opposed to the install details which is what I first thought of from "border") there are lots of pros and cons you can read, I think it's really based on style and personal preference.

Your first picture is that of a slight positive reveal which is the best for an undermounted sink. It makes for the tightest maintenance-free caulk joint between the sink flange and the bottom of the stone, the sink protects the bottom of the stone from chipping, and it's the easiest to clean. Plus it looks great.

The second picture is that of a top mount sink. It's easier to do so it's cheaper, you'll never get chipped edges, but every time you wipe your countertop you'll run into the sink flange which makes cleaning more difficult.

Flush reveals create an elongated silicone joint that frays and becomes contaminated and ugly.

Negative reveals don't protect the bottom of the stone and create a ledge under which crud can hide, however, fabricators love them because they don't have to be as accurate with their sink centers at template.

Your second picture was swiped from Simply Sinks out of Atlanta, GA. I gave Mike Williams life in the sink replacement business there 5 years ago. We stay in touch.