Growing up, I always had a couple Jennifers in my class. Both were called "Jennifer", but we had to add the first initial of their last names Jennifer M and Jennifer H). UKDaisy, about why people are hearing "Gin" when you say "Jen" - the only thing I can think of is that you might have an accent. I see you are from/in Kentucky. I noticed that people from your region have a strong slant on words like kin/been/pen etc. I can definitely imagine hearing someone from Kentucky saying "Jen" and having it sound exactly like "Gin".
As for me, I often have people confuse my name for Helen or Ellen.
.....Kelly