Nursing vs. not nursing really doesn't have anything to do with "independence." Nursing is just one way of having closeness, and we certainly wouldn't think that our spouce is not "independent" just because he wants some physical contact. So really, if you aren't feeling that you want or "need" to wean for your own sanity or personal space, I would encourage you not to push it. I know when I was pregnant the first time I thought nursing past a year was "odd," but then I nursed my first daughter for 14.5 months...and that was the shortest I nursed any of my 4 children.
At 13 months old, its not like you can really "reason" with your child about weaning--I know that some moms when they push weaning will put bandaids on their nipples and explain to the child that the "nursees need a rest," but as I said...13 months is too young for this to make sense to your daughter.
What I would encourage you to do is to stop offering a bottle/sippy cup as a "replacement," because while nursing is about nutrition, it doesn't sound like that is what it is about to your daughter right now. It is more about nurture for her. So try to replace nursing with activities to be close to each other. If you know about what time she would expect to nurse, a bit before that time start playing a game, coloring, playing with Play Doh, or something else to have some "mommy and me" time. You might try rocking in a rocking chair, although that might cause your daughter to want to nurse.
Good luck!