As a child, I stuck to a rigid rule - no one was allowed to call me anything but "Jennifer". Absolutely positively, I was Jennifer. Then, in high school, and certainly by college, I was going by Jen. Never ever ever ever was anyone allowed to call me Jenny. My high school tennis coach sometimes called me Jenny and it bothered me no end for a while, I told him off a few times, and then I just gave up. I saw him over the weekend this past week, for the first time in years, and he did not call me Jenny, but it's the first time I've spoken to him as an "adult". And I was struck by how much he reminded me of someone in my life currently. A few hours later, I figured it out. He was reminding me of Allan, one of my senseis. Allan, too, rarely calls me Jen. Recently, he's taken to calling me Jenster. No one else could get away with that. But I'll take Jenny from my high school coach and I can live with Jenster from Allan. I think for both of them, the nicknames are terms of endearment.
Now, I try not to let people call me Jennifer unless they're my parents or other family, still obeying the "you will call me Jennifer and nothing else" rule. But I introduce myself over the phone as Jennifer. More syllables makes for easier first time hearing. Otherwise people think I'm saying Jean, Jan, Gina, a whole host of other things.
Then there was the time at Starbucks when they asked for my name and when they wrote Jen on the cup and the pen slipped, it looked like they'd written Jew! Didn't think I would be outed at a Starbucks, but...