In the row settings panel under Visibility we have the option to either Match Site Sharing or Limit to Group. This is what "Limit to Group" is supposed to mean according to the note in the row settings panel: "Only those groups you select, organization administrators, and the site owner will be able to see this row on the published site." In other words, this row setting is supposed to override the site sharing.
In testing this capability in a couple of our sites, I find that it works for public viewing -- I can make a row in a publicly shared site invisible to the public. But the same row IS visible to any org member who is signed in to the org, regardless of the fact that the member is not a member of a group I've selected (in fact I have not selected any group to make the row visible to), an org admin, or the site owner. Am I missing something in how I interpret the instruction, or is this a bug? The behavior I am seeing is actually what is being asked for in this posted idea.