Bine is one of those words that has a rather indeterminate meaning.
Historically, its meaning was quite clear: bine was a variation of the word 'bind', and was used to name a number of plants: woodbine, bineweed, and simply bine. This last one is the plant that we currently call hops, and from which we make beer. There were a number of different types of hops, perhaps the most useful being the hoppan bine (literally, "climbing bind"); this is where the word hops comes from. We still use the old names for various varieties of hops, such as white bine and red bine.
At some point, someone decided that the 'hop bine' was a specific and intentional distinction -- that a 'bine' was clearly not a 'vine', but something else altogether. This was probably helped along by the fact that the word bine was also used in some cases to refer to any flexible shoot or stem of budding plants. Regardless, at some point someone noticed that hops vines were different from standard vines, in that they did not climb using tendrils or grasping shoots to attach themselves, but rather by twining around the object that they were climbing. This has since become a common definition of bine, and this is the definition that appears in Webster 1913, Wikipedia, and a number of other sources. As far as I can find, this is not a formal botanical term, but it is commonly used in gardening forums and among serious growers of hops.