I agree that the most relevant rating should be used.
Which rating to display
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It seems the rating shown when hovering a player's nickname is the highest rating that player has. This is not always the most relevant rating. I have an Online rating of 1346 after winning my one game, but a Live (Standard) rating of less than 900 after some 30 games.
I notice that the Glicko RD is calculated. Why not show the rating with the lowest RD? This should be the most reliable rating and thus also the most relevant.