Of course you are not picky! It is part of your identity, and as far as I'm concerned as different as Luci and Zelda, but more understandable.
As long as you are polite and gentle, she must be corrected, especially as she will be a member of your family. I think your offhand way of stating it was both.
I know a Bill who is Wilburn, not William. Ed can be Edward, Eduardo, Edmund and some other names. We had a whole running thread, or sidetracked thread, on that a few years ago.
My dad got in trouble with the US Army because he was taught to spell his name 'Glenn' and didn't even know until he was an adult that his birth certificate said 'Glen'. That's an extreme case, but shows the importance.