See also: Twelve Virtues of Rationality, The Meditation on Curiosity, Use Curiosity
What would it look like if someone was truly curious — if they actually wanted true beliefs? Not someone who wanted to feel like they sought the truth, or to feel their beliefs were justified. Not someone who wanted to signal a desire for true beliefs. No: someone who really wanted true beliefs. What would that look like?
A truly curious person would seek to understand the world as broadly and deeply as possible. They would study the humanities but especially math and the sciences. They would study logic, probability theory, argument, scientific method, and other core tools of truth-seeking. They would inquire into epistemology, the study of knowing. They would study artificial intelligence to learn the algorithms, the math, the laws of how an ideal agent would acquire true beliefs. They would study modern psychology and neuroscience to learn how their brain acquires beliefs, and how those processes depart from ideal truth-seeking processes. And they would study how to minimize their thinking errors.
They would practice truth-seeking skills as a musician practices playing her instrument. They would practice "debiasing" techniques for reducing common thinking errors. They would seek out contexts known to make truth-seeking more successful. They would ask others to help them on their journey. They would ask to be held accountable.
They would cultivate that burning itch to know. They would admit their ignorance but seek to destroy it.
They would be precise, not vague. They would be clear, not obscurantist.
They would not flinch away from experiences that might destroy their beliefs. They would train their emotions to fit the facts.
They would update their beliefs quickly. They would resist the human impulse to rationalize.
But even all this could merely be a signaling game to increase their status in a group that rewards the appearance of curiosity. Thus, the final test for genuine curiosity is behavioral change. You would find a genuinely curious person studying and learning. You would find them practicing the skills of truth-seeking. You wouldn't merely find them saying, "Okay, I'm updating my belief about that" — you would also find them making decisions consistent with their new belief and inconsistent with their former belief.
Every week I talk to people who say they are trying to figure out the truth about something. When I ask them a few questions about it, I often learn that they know almost nothing of logic, probability theory, argument, scientific method, epistemology, artificial intelligence, human cognitive science, or debiasing techniques. They do not regularly practice the skills of truth-seeking. They don't seem to say "oops" very often, and they change their behavior even less often. I conclude that they probably want to feel they are truth-seeking, or they want to signal a desire for truth-seeking, or they might even self-deceivingly "believe" that they place a high value on knowing the truth. But their actions show that they aren't trying very hard to have true beliefs.
Dare I say it? Few people look like they really want true beliefs.
I'm not sure there's an overarching "curiosity" that people have or don't have: I'm very curious about whether a specific kind of database will perform adequately in certain circumstances (long story) but I'm only mildly curious about how to identify which French painter during the 19th century painted which picture. Some art experts, I'm sure, have cultivated the skill to guess within seconds which painter it is for every picture. I wouldn't mind having that skill -- it sounds like a fun skill to have -- but it seems like it would be more resources than it's worth. OTOH, I really want my probability estimations re: the database to reflect reality. Do I need to use AI theory? Doubtful. Probably a little bit of statistics, and even that fairly mild, but I do have to think a lot about how to use my knowledge of databases to design experiments to find the truth out. I'm not sure if that would look "curious" to the lay person (and, of course, there's also a factor of "signaling curiosity" -- I want to make sure that everyone with a stake in the process sees that I've done the due diligence), but nonetheless, I'm truly curious about this (and yes, it could go both ways...I think this is the most important part of curiosity vs. fake curiosity).
When I was genuinely curious about the US immigration laws applied to me (and again, it could have gone both ways -- before running any experiments, I made sure to visualize both options, and realizing I can live with both) I just called an immigration lawyer (and, for a latter question, the paralegal who was involved with my visa). In that case I needed very little knowledge from LW -- I didn't apply my knowledge about Bayes, or about heuristics and biases, and just went and asked a professional (of course, in some cases, like wanting to know if a stock will go up, asking a professional is disasterous, but with immigration law, lawyers can estimate probabilities fairly accurately even if they lack formal rationality training).
Those were the two examples of real curiosity from my life that I could think of, that looked nothing like the description here of "real curiosity"....
I think this is a question of what satisfies your curiosity. Neither of the examples you give is a paradigm example of curiosity as such - curiosity is generally taken to mean a desire for knowledge for its own sake whereas both your examples involve seeking knowledge for practical reasons - but perhaps in your case these work and personal issues are enough to satisfy your curiosity. The fact that you're here makes me think otherwise though. Surely you read LessWrong out of curiosity?