Leo is not a nickname, though it can be used as such.
My grandson is called
Leo, but his full name is
Leonidas. My daughter wanted to use just
Leo, but my then son-in-law wanted it to be a nickname. She suggested
Leopold, he wanted
Leonidas, and he won the day. Not that my daughter doesn't love
Leonidas, she does. But she is like me----no nicknames as full names, so if she initially wanted to use just
Leo, you can rest assured that it's not just a nickname, but a full name in its own right.
I used to think that
Lucy was a nickname, because it can be used as one for
Lucille, but it was pointed out to me here on BtN that it is a name in its own right as well. So I've been set straight and can feel free to love
Lucy as a full name.
I think
Leo works as a nickname for
Leonardo, but it's too much of a stretch for
Lionel. There's also
Leopold and
Leonidas. But really if you like
Leo, there's no impediment to just using
Leo, even if you are a "no nickname on the birth certificate" purist like me.