Based on your clarifying comments, I come to the same conclusion that you have...it's the concrete slab that's the issue. By the sounds of it, it's actually routing water towards your foundation.
I had this exact same problem at my previous house. One side of the house had a concrete slab. Unfortunately, whoever poured it did a terrible job. By the time we purchased the house, the slab was sloped towards the foundation. Someone had previously tried to remedy this by applying a leveling compound on top that was now flaking off. As it was a monolithic slab (and was in MN) it also had huge cracks all over the place.
Whenever we got a severe downpour, we'd get a few puddles in the basement coming in from the footer. I purchased quite a bit of concrete crack repair (caulk) and filled in as many of the cracks as I could and all along the joint between the concrete patio and the foundation. That'd work for a season or two but inevitably the seasonal freeze/thaw cycles would open everything up again and I'd have to get more patch.
As we wanted to finish the basement, the eventual fix was for us to tear out the entire concrete patio and replace it.
I had a team come in with jack hammers and skid steers to remove the slab. I then brought in 4" of crushed rock and compacted the based ensuring I had significant slope AWAY from the house. I then put down 2" of sand, concrete pavers, and polymeric joint sand.
This fixed our water problems completely. I may have overcorrected the slope (the patio was noticeably sloped but I figured it was better safe than sorry. Our eventual paver patio ended up being about 10' out from the house, and slopped probably 8" from one end to the other. Never had water leak in that side of the house again.
Unfortunately, that is neither a cheap nor easy solution, but will say the pavers looked a lot better than the concrete in the end. The primary reason we went with pavers (aside from looks) is that we felt it'd survive the MN winters better and is a whole lot easier to repair if it ever needs to be. While there was a bit of settling here and there and the occasional weed or two you need to pull out of the cracks, for the most part, I was impressed at how maintenance free it was.
Best Answer
For a complete repair there are 2 obvious directions to cover which you mentioned. I'll go over what I would do in your situation.
Fix from inside
One product I would use is a hydraulic cement which can be found easily at your local home improvement store. This stuff is great for fixing large leaks in the basement.... if you can find the source. Hydraulic cement would be more permanent than a concrete filler or sealant and is commonly rated to 2000lb to 3500lb of hydrostatic pressure. Stay away from your cheap concrete sealant you would find somewhere like Home Depot. Most of these are water resistant, but not at all made for stopping leaks. Assuming you do find your leak source and fill every hole, the main cause is still unaddressed. For that we go outside.
Fix from outside
Fixing grading and downspouts will help a lot, however, in torrential downpour you may still get heavily saturated earth around your foundation, which is a bad thing. Even worse, (depending on where you live) is water expansion as it freezes in the cracks. This could seriously worsen the issue.
Probably one of the best (and probably the hardest solutions) is to use a polyethylene foundation wrap, something like this, in conjunction with a proper drain system at the bottom of the footing. Below is an example installation from the DMX installation guide found here.
A perforated drain pipe, crush rock, and proper drain or drainage field is necessary to keep the water moving. Below is an example of a proper drainage system as provided by the EPA
Hope this helps, and best of luck!