This happens in work ALL the time. Let's say I am called Elizabeth, but Liz to my friends / close work colleagues. I have no problem with being called Liz in conversation, but it's not what I go by in general and I always introduce myself / sign off my emails as Elizabeth, particularly when it's in a work context / introducing myself to clients.
I get so many emails from people I've never interacted with before starting with 'Hi Liz', or someone puts my name down as an action owner as 'Liz Smith'. It really riles me.
Whenever I email someone for the first time, I always use their first/full name as per their email address, then if they reply to me signing off with a using a shorter name I will use that going forward, but I'd never go straight in with 'Hi Jan' when the person's name is Janet.
AIBU?!
And yes I know I have too much time on my hands if this is all I have to worry about.....