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Both! The peak suction at the diffuser apex is an additional downforce contribution, but even with a really large radius there you will still have more massflow through the underbody and hence the high speed/low pressure situation.
On race cars where either the regulations aren't too strict or where other considerations reduce the importance of diffusers you sometimes see a quite gradual blending of the flat part of the floor into the diffuser. One of the highest-downforce car's i've ever tested was actually a road car but with diffuser channels running almost the entire length of the car, from a narrow 'throat' between the front wheels to twin 500mm square tunnel outlets at the rear. The only thing i've tested that had more downforce was a full-on ground effect F1 car with completely sealing side-skirts, and even that wasn't much better.
Not sure what your lecturer means. You can sometimes get a separation from the ground just behind a car with a diffuser. This is down to a boundary layer forming on the ground as well as the car because the flow under the car is faster than the road speed of the vehicle. This b.l is then still subject to the influences of pressure gradients and a combination of the upsweep from the diffuser and the return to freestream pressure from the low pressure under the car causes it to separate. Other than that, i've never known a diffuser be 'choked' by viscous effects unless it's at something daft like 2-3mm ride height. Not a realistic condition for anything with any suspension travel/compliance, though someone did once tell me that they run their F3 car at a ride height of -3mm, so the underfloor plank would be ground away down to the limit of the regulations and so get their car's underbody lower than was really intended by the rules. Interesting...
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