It's not really analogous to someone mistakenly using the "short" form of a name rather than the "long" form i.e. assuming that someone named Catherine goes by Cathy. That needs to be corrected right away, otherwise the person will probably continue calling her Cathy. But someone who mistakenly uses Patricia rather than Patty on a formal document is highly unlikely to switch to calling her Patricia all the time - that's the difference.
I wouldn't agree. I know more than one person who switched from calling someone whose given name IS the nickname to calling the person the more formal or longer form, e.g. calling someone "John" whose name is "Johnny" or calling someone "Jennifer" who goes by "Ginny" -- so not even short for "Jennifer" but "Virginia"). In fact, I'd be one of those individuals more likely to switch to calling someone the longer or more formal name than the nickname (if it truly were a nickname).
In the OP's case, it's not a nickname; she's being addressed by a name that is not hers. I get addressed often enough by a name I don't use (my first name; I use my middle name). I will correct someone right away on that because I don't think it's a good idea to let the person continue to err then finally decide at some point later to correct the person. Obviously, YMMV in this case, but I figure it falls on the person whose name is being stated incorrectly.
As far as being "too picky" by correcting her FSIL via e-mail, I think you're using the wrong phrase. It sounds like you mean "impolite" when saying OP should've made the correction via a phone call or in person. That's not being too picky at all.
Also, I think you're missing the point that this isn't simply the case of being called the formal longer name; she's being called the wrong name, period, regardless of whether or not her legal name is normally recognized as the nickname of another name.