Bike agility and handling

This has zero to do with tyres.

The answer is frame geometry (wheelbase length, head tube angle, bottom braket height probably being the main factors).

Don't ask me to explain it.. but I know exactly what you're talking about, my TCR can be flung around corners like an agile cat, however my defintion sort of glides around corners like a swan and needs more work to get around tighter corners at speed. The definition has a slightly longer wheelbase to the TCR and I think a lower BB also.

The Supersix Evo is known to have a slightly more relaxed geometry than other bikes.

Tyres? I've had those symptoms on motorcycles with tyres on the way out. Also a horrible high speed vagueness on a bicycle that I spent ages trying to find, which turned out to be caused by a badly patched tube on the rear.

So it might be worth trying a tyre switch.

When you say same fit, does that mean same stem length and bar reach on both bikes, or are they different in order to hit the same fit coordinates. Without knowing the full geometry of both bikes it's hard to say what is causing the difference, but most likely it's the combination of trail, front centre and body weight distribution.