Since this is a game about cognitive biases, hosted at mit.edu, I maintained hope that they, too, would be this pedantic about semantics, but, sadly, no.
Another example that I would say 0 to is, how many times did Jesus Christ come back to life? But I would say 1 to, according to the Bible, how many times did Jesus Christ come back to life? The difference between Star Wars and the bible is that there are not large groups of people who insist that Star Wars is real, so I feel safe in assuming an implicit, "According to this fictional thing". I consider the "reincarnations of the Dalai Lama" to fall under the bible category.
Or, perhaps you're implicitly talking about the world in which the Dalai Lama is reincarnated. There's not much difference to your distinction.
The correct answer is whatever is commonly excepted by Buddhists, I guess, or perhaps there is an official number. 0 would be a trick answer.
It's also very disappointing to have the site claim that there is one unique "right answer" to questions that are ambiguous. For example, the site claims that 30 mph is the "right answer" to "how fast can a race horse run", but Googling shows plenty of examples of race horses running 40 mph or faster. I think they meant something more like "what is the average speed of race horses when clocked during races", but that's not what the question said.
I answered 13 since the current Dalai Lama is the 14th and the first wasn't reborn. Also, to get technical he wasn't reincarnated, but reborn, which is an actual distinction Buddhists make.
Kidding aside, I get your frustration. Trivia questions by necessity have the format of "one short question" = "one short specific answer".
Real facts usually don't work this way. Historians argue or change the date of various historical events over time as more data emerges. The Solar System "lost" a planet few years ago, and so on. "Facts" are in constant flux.
There's also a complicated, detailed context, there are a lot of "depends" in a real answer. Say, "how fast a horse runs"... domesticated & trained or wild, what breed, on a road or on a meadow, with horseshoes or without, etc.
So when faced with trivia questions the last thing I could say is I feel "overconfident" answering. Especially when the answer is a number. The designated correct answer is just an arbitrary data point that the author stumbled upon earlier in a book or online and wrote down as the answer.
It's apocrypha masquerading as facts.