I think for me, this issue in part is because its your nickname. If they were misspelling your actual name, well then that would be straight up wrong - you no doubt have official, legal, irrefutable proof of how your actual name is spelled. But your shortened nickname is merely your preferred spelling. "Anny" and "Annie" are the same nickname, they are simply different spellings, but in either case your name is actually Annabeth. This happens with many names: my friend Jennifer has had plenty of people shorten her name, to Jenn, despite the fact that she shortens it to Jen (or goes by full name).
I think when its not your official name, but rather a nickname, and you approve the nickname (vs always going by Annabeth and never using any variation or shortening) its more of a pet peeve then actual disrespect. In this case its totally a benign difference so sure it seems reasonable a person should go with your preference, but where's the line? What if you wanted your nickname to be spelled AnNyeee? Its still the same verbal nickname... but kind of a hassle to type out and remember what with the capital in the middle and 3 "e" in a row. Should all personal preference variations of nicknames be observed, in all cases (such as work email)? And if there is a line about whats reasonable and whats not, who decides where that line is?
In my opinion actual names have concrete correct/incorrect spellings. But nicknames... I will try my best, but not get too wrapped up; if I mess up, well the person who voluntarily chose that nickname needs to understand they choose the odd variation instead of the common one, and the consequence of that choice is people forget or get lazy.