The drain would be surface level but graded on both sides. You can do custom, custom order, or off-the-shelf. Theirs was custom order since it was so long. Here's some internet pics to give you an idea:
The above would not replace the foundation drainage but augment it.
Since your foundation is acting as a dam, it doesn't take much rain to create a large amount of water in your basement. At 1 inch of rain, a 1,000 square foot area receives 600 gallons of water.
I'm in a dry climate so a big summer rainstorm is ~1/4 inch. That equates to ~540 gallons of water deposited within the immediate drainage footprint of my house (house area + 10 feet out from foundations). I know this is fairly accurate because I have 6 @ 55 gallon rain barrels on my downspouts and they fill & overflow with one of these rains.
As far as how much of that water soaks into the ground... You said the water in your basement rises when it rains. Water follows the the easiest path which right now is against your foundation & into your basement. Grading & drainage will change that so the easiest path is away from the foundation. Grade causes water to go out out before it goes down...