Gardiner could well have been "gardener". However Bingley is a place, not a profession. The standard rule (for what it's worth, and it really isn't worth much)is that if your name is a "place" name, rather than a "profession" name, ora an "attribute "name like White, Long, etc. then it's a sign that you have an ancestor that was Lord of the Manor.

Whether Jane meant anything by it is pure guesswork.