Looks like it might be on a piece of the protective coating. However, I didn't know the 16610 came in a version with a laser etched ring.
When was the Rolex engraved rehaut introduced?
The time of which the engraved rehaut was introduced varies between models, as Rolex slowly but surely introduced it for all their models.
In the beginning, Rolex introduced the engraving on many solid gold models, however, the engraving was first introduced on the Turn-o-graph around 2004.
The Rolex GMT-Master II got engraved rehaut in 2005. This is at the same time as Rolex updated the GMT-Master II model. The new design was initially exclusive to the all-gold version of the watch (ref. 116718); however, the new model featured a number of technical changes. In 2007, the rest of the models got it as well. Sports watches such as the Submariners, Explorer II, Yacht-Master, and Daytona were updated half through the Z serial in 2007.
The Daytona in stainless steel got it around 2007 with the Z-series. Same applies to the Explorer II 16750.
The GMT-Master II reference 16710 and Rolex Explorer 14270 never got the engraved rehaut.
The last of the Explorer 114270 did get the inner engraving. This was near the end of the model run, before Basel when the new one was introduced:
Most z serial Rolex watches do not have the inner bezel engraving.
D and F serials overlap and either could have it, just as the Z and M overlap with the rehaut on the Stainless steel models.
All the 116400 Milgauss have the engraved rehaut.
All white gold Daytonas on Oyster bracelet have the engraved rehaut. This particular model was introduced at Basel 2004.