The second option is to just have "hidden" on the leaderboard, you can click on hide me again like the other one, it actually shows you the real top 10, but some of those names are censored as "hidden name" so you will not know who it is. Those are two very different approaches, I don't know which one I would prefer more, but I think the first one is closer to my heart.
The screenshot of me is from a leaderboard of a wagering contest. There the first option you suggested would not make sense, because the user would not appear on the leaderboard and would not be eligible for a prize. Participation in the contest would not make sense at all if you don't want to show up.
Option 1 would only be possible for weekly/monthly/yearly Leaderboards. But even there, the results will be distorted if entries are missing and you can never be sure if you are the weekly leader or if there are not 10 players in front of you, but they didn't want to appear in the list.
That is the thing though, if you hide the name, it was a discussion at other places as well and have been a point of argument between a lot of people. If you hide the name, how can we really know its a real player or just the house playing against itself? A house could basically play 100 btc in an hour without worrying and just lose only a few.
In my opinion this has nothing to do with whether the name is displayed or not. For example, you don't know if there is actually a person behind the player "bestgambler123", someone just runs a script or the casino plays against itself. So it doesn't matter if it says "bestgambler123" or "hidden", it has the same effect.