I'd say that Leo is a name in it's own right but it is also used when shortening other names, so she wasn't entirely unreasonable to ask if it was short for anything else as long as she wasn't insisting that it should be.
My name is a name in it's own right but it can, if you try quite hard, be used as a shortening or pet name for other, longer names. It's rare than anyone assumes I have one of those other names, there's a very famous person with my name and so it's not like nobody has ever heard of it as a name in it's own right. Most people with my name are not called something else.
But I did spend two years of school trying to convince one teacher my name was not Geraldine because of her assumption that my short but very real name was a shortened version of a name that sounds nothing like it 
As a joke at work once, we referred to a customer who goes by the name of Mick as Mickolas (it was Christmas and he pretends to be a bit 'bah humbug' about it but secretly enjoys it so we were playing on Saint Nicholas) and now one person working there actually believes that Mickolas is his real, full name 