It is based on conjecture and guessing a lot more than on hard evidence, so be warned, it's kinda headcanon-y.
Pink Diamond is a very, very weird gem all around.
1. Her color isn't a primary color - unlike Yellow and Blue, she's off this primary color axis.
2. She's immensely, unreasonably powerful, way more so than other Diamonds, but only on rare outbursts of emotion when she goes REEE.
3. She was not at all prepared to be a ruler, as opposed to the way Gems normally come out - ready for their job, even if a bit naive and silly.
4. White, paraphrased: "You were always a part of me I had to repress"
A common theory for the origin of Yellow, Blue and Pink is that White split off bits of herself to keep herself pure, without problems. Here's how I think it went.
She split off sadness and excessive emotions Blue;
She split off warmongering as a form of coping with stress and unhealthy bottling of emotions instead of discarding them into Yellow;
And, finally, she split off her fury, her rage, her excessive passion into another Diamond, Red Diamond. Buuut she couldn't go through with it. Be it because she physically couldn't, messed up the process, or whatever may be the cause, she ended up splitting off just a part of herself that embodied passion, keeping a bit of it in her - a bit that came to the surface during CYM. It wasn't just a blush, it was off-coloration. Gems can blush; that's normal. Gems do that.
White specifically went off-color, pink. She would've gone red off-color, but what do you get when you mix red and white?