If Mech1 puts in a drainage trench across the back of his house, from higher to lower on the property, deep enough to be lower than the floor of his basement, stone and pipe with cover blockage for material intervention, why wouldn't that take most of the water away... ? i.e. path of least resistance !
I had a similar water problem, but not as bad as Mech1's, during the spring thaws each year my crawl space 20 x 50 x 4 feet deep, would almost fill with water. Thanks to only the fact that one end was higher, where my furnace was, did I not lose the furnace, hot water heater and maybe my fuel tanks. I couldn't find the problem and pumped every spring with a small sump pump took enough away to save the important stuff but it was getting musty smelling.
I already had enough stone under the house and sides with piping, so I decided maybe it was the gutters, so I started digging the furthest away point, dug approximately two to three feet where I could, and WHAM-BAN, running water.
I am at 70 feet above sea level, on a slight high point of a downward hill with a very deep gorge on the back side of my house and enough stone under the house and floor to sink a battle ship, and I was confused as to how water could flow from the gorge side at three feet down... but it seem that it might be the problem.
I dug to three + feet, did the stone and pipe thing, covered it and Thank you God, no more water.
Even though the water flowed in that ares for who know how long and under my house for at least 10 years causing me a PITA situation, it was gone, and a little too easy but it is what it is.
So I keep saying the same thing, find the source/flow of the water, if you can dig a trench deep enough and see the water flow, then you should be able to divert some of it, maybe all of it.
Because of the volume of water being tossed around on this thread, I am thinking it may take more than what I did but it is possible to divert Mother Nature.
From the pictures, the backside of the house has higher land than the front or side, the backside needs to be excavated and determine what is going on. If the water IS coming UP instead of from the side, yes there is an even bigger problem but if you can get flow in a trench then you can do something about it.
The idea of filling in the basement... interesting, a little off the wall, but a lot of crushed stone, and a thick concrete cap, with pipes taking the overflow out one side to a tank system, where you can pump water for whatever you want... and if there is enough... you can even sell it until the owner of the Pond that is draining through your yard complains he isn't getting his share... LoL !
Mech1, Good Luck, this is a good hydraulics problem, another idea maybe but just thinking out loud, a local University can help, some school are always looking for a problem to solve.