Hmmm no, to me a short form isn't a nickname. A nickname is something like
Abby for
Abigail or
Maddie for
Madeline but nothing that isn't used as a nickname anymore and has a long history of a name on its own. It simply derived from that name.
I used to live in a country in which
Annika was one of the most common names around. No one there ever used it as a short form of
Anne or
Anna. It may have derived from those names but always stood on its own. I would never say that
Annika was a nickname.