Identify this 12 wire cable
It is probably 6-pair phone cable. Based on the picture, this is almost certainly not CAT-3, as the pairs are not twisted together.
The canonical example of such wire was 25-pair, used throughout phone systems of yesteryear. The catch is that old phone systems used 1 pair for each simultaneous voice connection and then a number of pairs (how it worked varied quite a bit) for very low-speed signaling between the phones and the main unit. If it turns out to actually be CAT-3 then you should be able to get 10 Megabits out of it. If it isn't even CAT-3 then there are various adapters, line drivers, etc. and you should be able to get a couple of Megabits (i.e., similar to DSL speed, though different technology), but not necessarily much better (and possibly worse) than the coax you have now.