Of course people are perfectly capable of being wrong about something and saying it's their opinion doesn't change that. However, it's often difficult to tell what the truth of a situation is. You may think someone else is wrong when they're not. Therefore any given person telling you that you're wrong isn't necessarily an indicator that you're wrong or that they have better sources than you. Even if you are right, that doesn't necessarily mean you're convincing. Even if you're convincing that doesn't necessarily mean that you've presented a justifiable argument. Even if you've presented a convincing and justifiable argument they may disagree with you. Not everyone is convinced by the same arguments, and sometimes even an argument that seems convincing can be defeated by a more strongly held sense of knowing the truth, whatever that's derived from. There are a couple other things around this concept that I think are also important to address, because they come up a lot. Not agreeing with someone once you've heard their argument is not the same as not understanding that argument. I'd say that we'd be almost universally better off attempting to determine why it is we disagree than simply insisting that our perspective is true and trying to prove this. Otherwise, how will we ever know if the other person has a piece of the puzzle that we're missing? Unless we're so arrogant as to assume that we know everything, in which case we most assuredly know nothing of significant value, we shouldn't jump to the assumption that disagreement means the other person is wrong, we should use it to investigate the accuracy of our own perspectives. Sometimes we're clearly right (though bias can seem quite clear), in which case the best we can do is offer a learning opportunity. This is unlikely to be achieved without being willing to understand the perspective of the person who is lacking or rejecting information. It certainly won't be done by waving away their perspective as if it didn't exist. None of us are the arbiters of truth, but none of us are the arbiters of truth.
I don't really think this is about the edge cases, though. This is about the people that respond to the geological record with "Well, that's just your opinion." It's a thought-stopping technique, a way of not thinking about evidence that might contradict an unfounded belief/opinion. I definitely agree there's subtlety and nuance to the word and the idea of subjective opinion in proper use - but the article is about whacking people who aren't using it properly to begin with, and not being inappropriately courteous in letting those moments pass unremarked.
Sure, but in the process of that we have to make sure we're not simply assuming we're right where we're not. What someone's really saying when they say "that's your opinion" or "that's my opinion" is that they don't consider your argument compelling and aren't interested in continuing the argument. You're right that it's a conversation ender, but people have a right to end conversations. Maybe to you that conversation sounds like you trying to explain that 2+2=4, but to them it sounds like you won't shut up about 9/11 being an inside job. Perspectives aren't absolute even if truth is. We should try to remember that and draw humility from it.
Oh, I hang in a state of constant oscillation between arrogance and self-doubt. But in general, I prefer to be correct and let "right" sort itself out. But yes, I do get exceptionally annoyed with people who go on and on about things like ... well, 9/11 being an inside job. I consider that one of a range of possibilities - it's clear people had to conspire to make what happened happened - I don't feel that we can be sure about this. It's kind of like people who talk about UFO's coming from Venus. "When you are unclear about what UFO means, we don't really need to talk about your grasp of the habitability of Venus, do we?" "Extraordinary claims require extraordinary evidence"; "assertions made without evidence may be dismissed without proof." There's a considerable number of tools to filter out the bullshit you don't need to deal with. Not wanting to deal with the remainder, though - well, you may have every right to not deal with it - but not dealing it is rarely the most productive course.