In my opinion, when there's mist or very light droplets falling from the sky, it's not raining. It's raining when I need to wear a rain jacket to go from my house to my car. In my opinion, it only rained once this summer, where I live.

That's an opinion on the definition of "rain". Language is inherently subject to opinions, since it's descriptive, not prescriptive. Words mean what you want them to mean. It's through mutual agreement that we get a solid definition. But the definition is solely an opinion of what the word should mean. The fact of the matter is that there's mist or very light droplets falling from the sky. Your opinion is that you don't call that rain. That doesn't mean there isn't water falling from the sky.

Also, I'm allergic to nuts, so in my opinion, nuts are awful.

Your opinion is that nuts are 'bad' (a subjective opinion term) because you are allergic (a fact). This is an informed opinion. You don't like X because Y. Nuts are not objectively bad. Not everyone is allergic.

These opinions are obviously false from the consensus point of view, but they are still true from my point of view... That's the core of the problem.

That's called a subjective claim/assertion. Or a 'value judgement'. Ultimately it can only be verified from a subjective point of view, meaning that contradictory results are expected. They don't describe objective reality. They describe a subjective view of objective reality. They cannot be right or wrong, given that they are subjective.