S., your daughter's use of water brought a smile to my face. I guess she is experimenting on what happens when you open your mouth when it's full of water. At least all she gets is wet.
Pediatricians can be very calendar conscious, but kids progress at their own speed. If it takes until she is 13 months or 14 months to get totally on a cup, then it takes her that long. I wouldn't sweat it. Up to now, sucking is how you get something to drink, and water in the mouth is something to play with. She'll figure it out, but I don't think its bad behavior, I think it's more just kid behavior.
Kids are little scientists, they have to experiment over and over to make sure that their prior observations are correct. What if that toy fell UP the next time? Will the water stay in my mouth all by itself the next time? How far can I open my mouth before it falls out? These are questions that they need to learn, and they can only learn by testing it many times. After all, they are not born knowing the laws of physics, they learn them the same way we did, and we drove our mothers crazy in the process too.
So, as long as she is getting what she needs, and if you get an underpad to put under her chin when you give her the cup, eventually she will learn to use it. We all did, each in our own time. And tell your pediatrician that you are working on it, and she hasn't completed her experiments with fluid dynamics yet, lol.