Are you saying that "你要相信你自己" is grammatically incorrect?
I like your version, but I would have though it was okay with or without the second "你". Certainly "你自己" itself is an extremely common phrase, as far as I can tell.
Yes, it is technically incorrect but definitely commonly used. Many 语病 has its source in commonly used phrases, actually.
Why is 你需要相信你自己 wrong?
It's a bit redundant I guess? Tho I wouldn't mark it wrong, it reads more of you need to believe in yourself (which is stronger in nature)
No. That's "You yourself have to believe". (I.e., it's not enough for others to believe.)
But a close approximation is perhaps "你要有自信（心）", "You have to have self-confidence". Native speakers, is this correct?