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?
Yes, I do seek knowledge for other reasons, here and elsewhere. But my expectation that this will not "look like" curiosity because I expect to have few changes in my behavior based on what I read, and so the importance of it being "true" is likewise diminished. Sure, I would like to have my beliefs about the brain and AI be true, but I'm not prepared to spend a LOT of resources to do it -- I'm sure if I were really curious about the role of Oxytocin in relationships, I could reach true beliefs faster by spending more resources. There a...
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.