No offense intended, your insights have been razor sharp. Indeed, on several occasions I've thought, "wait, he's right... no wait, I'm right..." But I see it clearly now, and there is only one point that I am having trouble making to you. It looks like you and I are all that remain in this debate for the time being, so it's up to us to get to the bottom of it.

Agreed.
Also agreed. As long as they never use the information that they are the 10 billionth, your statement is completely correct. But there's something we must agree on. The only meaning to the statement that "I am 95% confident" of anything is that I have in mind a collection of cases, or trials, or individuals, that are effectively indistinguishable, except for some random stirring which is being examined in regard to the confidence interval. How are you going to even define your confidence interval, if you don't use this definition? You have in mind a set of all humans, and the stirring is the difference between you and someone else. So yes, of that set, 5% are in the first 5%, it's well defined. But the same must be true of any statement of confidence. Therefore, if you claim that "I am 95% confident that humanity will not outlive 200 billion", you must also produce such a set (even if idealized) or it is a meaningless statement. Note this statement sounds very similar to your statement above, but it is not the same, because it explicitly uses additional information-- I am the 10 billionth. The first statement has a set where 5% are wrong, the second set does not, not even in principle. This makes it an incorrect probability argument, much like the set of people who saw a $10 envelope and figured it would be the cheaper envelope half the time. How do you prove that claim wrong? By asking for a set of people seeing $10 and having it be the lower envelope half the time. Could the supporters of that probability view simply say...
as if it saved their argument? The fact that there is no such set is exactly why the argument is wrong.

Exactly. This is why you simply cannot be 95% confident that the 200 billion limit applies. The within species probability is not enough, because it is not a probability that tests a confidence interval about a population that makes it to 200 billion, it only tests a confidence interval about being in the first 5%. These are different tests! What do you mean by "I am 95% confident our species won't outlive 200 billion." Where is the 95% who are right about this exact statement, right from their own mouths, and the 5% who are wrong? Carter must change the statement to "I am 95% confident our species won't outlive 20*x, where x is my birth number". That statement only checks out if you range over all x, but then where is the 200 billion? This is just like you had to range over all y to say that the expectation of the second envelope is y, and you realized that isn't true for a given y=10. It is just the same thing here.