By exterior waterproofing, do you mean checking the drainage around the house? I was referring to measures taken on the exterior of the foundation. But, of course, diverting water well away from the foundation should help too. That could mean regrading. And the often overlooked movement of downspout water far away from the foundation--not just a couple of feet away on a splashpad. I also wonder too if the act of lowering the basement significantly increased the risk. Oh yes! The worst I ever personally saw was when renovators lowered a basement in a century home by less than two feet. Water started pouring in and before they could find and install heavy duty sump pumps, there was five feet of water in the basement. They had hit one of the many underground streams in a city (Toronto) basically built between two rivers running into Lake Ontario.

A French drain is any subsurface perforated drain pipe that is installed in gravel but in the US this term often refers to such a system installed on the interior of a foundation wall instead of the exterior presumably because only the French would put a drain on the wrong side of the wall. I assume the foundation, slab and backfill are already in place and that water is getting into the basement from more than the one hole and that the proposed drain will be cut into the basement slab and terminate in a sump pit where a pump can remove the water. If not, please be more specific about the situation. The design of subsurface drainage and waterproofing is normally the responsibility of the architect when one is involved. The soils engineer would be concerned with soil bearing capacity and the structural engineer would be concerned with the entire structure. However, in most jurisdictions anyone can design a house so it is up to the owner to assign responsibility for different aspects of the design. The bottom line is that the person who wrote the specification should have included a description of the subsurface drainage and waterproofing. Perhaps the architect and the engineer each thought the other would design the waterproofing and drainage system but to me it is very odd that the contractor and the building official didn't catch such an obvious omission. I specify spray-on waterproofing and footing drains for any basement regardless of the nature of the soil, site slope or surface drainage. It's simply cheap insurance. There are companies that specialize in cutting the slab back from the foundation wall and installing a perimeter drain which is a special kind of French drain since there is a slot left open along the wall to catch water from the wall as well as the ground. If there is a good layer of crushed stone under your slab, a sump pump in a pit might adequate.

Update: Last night, shut off valve fixed. I cut 2 small holes (enough to get my head and an arm in) in the ceiling in the basement and a smaller one in the wall. The first hole in the ceiling was in the floor space where the 2 recess lights are that the water was pouring out of. Sheet rock was completely dry, couldn't feel any dampness at all. Second hole in the ceiling was directly below the toilet and it was slightly damp but not bad at all. Have a fan up there now blowing. Smaller hole in wall toward the of the ceiling, no dampness and insulation dry. I honestly don't get it. Could it be that it was because it was such a short period of time that the water ran and it had a place to escape that it doesn't seem wet? Maybe it's because the other end of each of these spaces goes into my unfinished storage room which also gave the water another place to go? I have fans in there as well blowing up into the spaces now. It appears that I've caught a break (unless I'm missing something?)

Since it hasn't happened under similar circumstances before, it may well be a clog or obstruction in the drainage system. If you know where your drainage system runs to daylight (assuming it does), check the outlet pipe. It may be as simple as debris or sediment build-up in the end of the pipe. If you can't see any obstruction, or the system empties into a sewer, infiltration basin, etc., you may need to have a plumber or sewer/septic contractor come take a look. Find someone who can run a camera up the drain outlet and/or down the floor drain and inspect for blockages. Many such contractors can do at least a temporary repair without needing to dig anything up. Unfortunately this is one of those things that might be no big deal or might be more serious. Either way, good luck!