Neither is wrong. If you want "fuller grammar" you'd favour the first option; but when you're printing on a ring, since the context requires extreme simplicity, either way is fine. In fact, I would prefer "Born yyyy/mm/dd" if it were an option. Regarding dates, since people here are suggesting to use the American version (mm/dd/yyyy), I have to say it's an inverse to the British version (dd/mm/yyyy) and that creates a lot of confusion in our modern global world. The benefit of your version (yyyy/mm/dd) is it's universally intuitive since it's mathematically ordered - you can even add a time on the end and it stays in order from longest period to shortest period. This helps when you're sorting dates digitally. For this reason, your date format is very popular in computer programming, but it may still seem strange literature buffs. Give it a few decades and your version may be everyone's default.

In the U.S., we would write "Born on mm/dd/yyyy." In many other English-speaking countries, they would probably put the date ahead of the month, so: "Born on dd/mm/yyyy."

The top one, although in the u.s. the most common format is MM/DD/YYYY