Anthropic reasoning is any reasoning based on the fact that you (believe you) exist, and any condition necessary for you to reach that state, including suppositions about what such conditions include. It can be supplemented by observations of the world as it is.
In this problem, most of the problems that purport to use anthropic reasoning, and the original presumptuous philosopher problems, they are just reasoning from arbitrary givens, which don't even generalize to anthropic reasoning. Each time, someone is able to point out a problem isomorphic to the one given, but lacking a characteristically anthropic component to the reasoning.
Anthropic reasoning is simply not the same as "hey, what if someone did this to you, where these things had this frequency, what would you conclude upon seeing this?" That's just a normal inference problem.
Just to show that I'm being reasonable, here is what I would consider a real case of anthropic reasoning.
"I notice that I exist. The noticer seems to be the same as that which exists. So, whatever the computational process is for generating my observations must either permit self-reflection, or the thing I notice existing isn't really the same thing having these thoughts."
Each time, someone is able to point out a problem isomorphic to the one given, but lacking a characteristically anthropic component to the reasoning.
To me, that just indicates that anthropic reasoning is valid, or at least that what we're calling anthropic reasoning is valid.
One day, you and the presumptuous philosopher are walking along, arguing about the size of the universe, when suddenly Omega jumps out from behind a bush and knocks you both out with a crowbar. While you're unconscious, she builds two hotels, one with a million rooms, and one with just one room. Then she makes a million copies of both of you, sticks them all in rooms, and destroys the originals.
You wake up in a hotel room, in bed with the presumptuous philosopher, with a note on the table from Omega, explaining what she's done.
"Which hotel are we in, I wonder?" you ask.
"The big one, obviously" says the presumptuous philosopher. "Because of anthropic reasoning and all that. Million to one odds."
"Rubbish!" you scream. "Rubbish and poppycock! We're just as likely to be in any hotel omega builds, regardless of the number of observers in that hotel."
"Unless there are no observers, I assume you mean" says the presumptuous philosopher.
"Right, that's a special case where the number of observers in the hotel matters. But except for that it's totally irrelevant!"
"In that case," says the presumptuous philosopher, "I'll make a deal with you. We'll go outside and check, and if we're at the small hotel I'll give you ten bucks. If we're at the big hotel, I'll just smile smugly."
"Hah!" you say. "You just lost an expected five bucks, sucker!"
You run out of the room to find yourself in a huge, ten thousand story attrium, filled with throngs of yourselves and smug looking presumptuous philosophers.