Correctly tagging a new island

It is small, and not even shown in Google maps, but it does have a name. Where do I put the name? Just a “place”: “island” somewhere on that rock, and then setting the “name” attribute?

Yes, just draw in a polygon for the island and set place=island and the name attribute.

What do you think, should the island be marked as well natural=land, or alternatively as a hole in natural=water polygon relation in case of island in a lake, or perhaps natural=coastline if the island is in a see?

So far, I adopted the tag “natural=coastline” from the main island (both islands are in the ocean). Please lat me know if I need to change something.