Ihfarm,
Knowing what we know about your situation I don't think you have any other choice but to go with the sub-floor drainage system, which would be my first choice, but as you can see others believe as strongly in the exterior system.
You still need a sump pit for a junction, but might not need a pump if the side of your house that is on the down hill side is accessible (no additions, porches etc) use your back hoe to dig a trench to the area where the existing drain came to daylight, always better to drain by gravity than rely on a pump/electricity.
If you go with the gravity trench, use solid pipe out side and you can tie your leader pipes/surface drains into the same trench, just not the same pipe, you want to keep them separated and the outside drains don't have to be that deep in the ground as the one coming out from under the basement floor, keep a narrow trench and the outside drain can go on top of the basement drain.
You do need a little pitch to prevent sediment build up, but it's gonna happen anyway due to the slow passive movement of the water from under the house, that's why you need the junction pit so you can flush test/clean it from inside. you don't have to get carried away with the pitch, even 1-2 % is fine, but at no point can the pipe have a hump in it that is above the floor level in the basement and as the pipe gets in the frost zone you don't want any dips either, to prevent freezing. you want to set the discharge line so that the level of water under the floor is a min of 6-8 inches.
A simple way to check all these specs is when the trench is open before any pipe is put in, test it with water in the bottom of the trenches (inside and out), the bottom of the trench should be the exact contour you need and then any piping (outside) will be solid so you don't need any stone or filter fabric just lay the pipe on the bottom and back fill with dirt.
John.