It’s not connected to providence, because even according to your view, the Holy One, blessed be He, can intervene. What I mean by not wanting it is that I’m not happy when it rains where I live.

The question isn’t clear. Why does it matter whether you’re happy or not? There’s no need to feel joy in order to ask the Holy One, blessed be He, for things. The commandment is to ask for what the world needs, not to rejoice.

Because it’s a vain prayer, since you don’t really want rain at all.

Why is it a vain prayer? You asked for rain, and rain can in fact come down; there’s nothing “vain” about that.
And if it doesn’t make you happy, that’s not so terrible.
You can ask for things even without feeling a surge of joy in your heart.
We do this all the time in everyday life. (Every time you ask someone for something and they agree, are you flooded with joy?)
There’s no obligation to feel joy, even though it is certainly a virtue to rejoice in the good that comes to the world. All you really need to do is ask for what the world needs.
If you were claiming that, in your opinion, rain brings harm to the world and therefore one should not pray for it, that would be a different discussion. But when you admit that it is good for the world and only have trouble aligning your feelings with that understanding, that does not prevent the prayer.

Yishai, let’s think about it the other way around.
A person prays for something that he thinks he needs. He wants to be rich and win 50 million in the lottery. The Holy One, blessed be He, knows that this would be wealth kept for his harm, so his prayer is not accepted. The person himself is very happy as he prays.
Is that a vain prayer??
In other words, a person’s own thoughts have nothing to do with the definition of a vain prayer. He may be happy or sad. But in practice, he is praying and does not know whether his request is right or not, and he also does not know whether it will be accepted.
Of course it is better to pray joyfully, because what good does sadness do? And pity those who see only themselves in prayer and not others. Aren’t there other Jews who need rain?
In any case, the Talmud in tractate Berakhot, chapter 9, defined a vain prayer as a prayer about the past. And there is a dispute in Jewish law whether that means it is forbidden to pray or only that it is ineffective.

I defined it incorrectly. I think the problem is that a person says one thing with his mouth and another in his heart, in a case where he doesn’t really want it to rain in practice and yet still prays for it.

All right, first of all, I already wrote that this is not a vain prayer.
Second, let’s go with your approach.
Let’s say a person prays, “Heal us, O Lord,” but he wants his own illness to continue a little longer so that he won’t have to go to work, or so he can receive an allowance from National Insurance. Is that saying one thing with his mouth and another in his heart? No!
Because: (a) he also intends the healing of others; (b) he intends that he himself be healed when the right time comes and when necessary.
And based on that, you too do not mean that no rain should fall at all. Surely you don’t mind if it falls in other places, and surely you don’t mind if it falls for those who need it.
(And if you insist on asking about someone who does mind and still does not want rain in any case, I’ll answer jokingly that whoever keeps digging and digging becomes ridiculous. Good night.)