and what characteristics make it more "involving"? Less grip, more like it's about to slip and slide about under you?
There are many things that go towards an involving drive. Steering feel (Audi is poor at this, sadly BMW has never been that good but they have gotten worse) is important to many. This means the steering feels more connected to what the road is doing, it provides delicacy and you can't really understand that if you have never driven a car with exceptional feedback such as a 964, 996 GT3, Noble M400, Caterham or a well set up MX5 (not a standard one as they are far from optimised). Then you have balance and how the car deals with weight, BMW is usually better at this than Audi as they have focused on the weight distribution more. Then there is braking feel, how much feel does the car give you into a bend or on different surfaces, something Audi is mostly crap at. Then you have damping which is how well the car handles the road undulations, cambers, bumps, surface changes. Audi again has a habbit of building for German roads, super smooth often and when put on bumpy British roads it starts to go wrong. It's the balance of all of these elements that provide enjoyable feedback and the driven wheels have a significant impact on this as front drive (be it 2WD or 4WD) will corrupt that over a car that just steers and suspends at the front.
Audi makes safe cars, but the reason they constantly get a hard time (outside of things like my R8 which to quote Harris is "so unlike and Audi") in the motoring press is most of those drivers get to drive all cars so know what good can be. If you don't then you won't care or know or if you don't think those elements are important you are not someone who drives for the sheer challenge of it. Nothing wrong with that, but to me I found the RS3 a fast, nice looking (saloon) well put together place to be, but a somewhat detached and blunt drive, if fast and nice sounding.
Chose the bits important to you, buy the car you want. Simple, no right and wrong, but facts are facts.