A max rate turn is not flown at the minimum, but the maximum possible speed. Basically you need the maximum possble lift, inclined at the maximum possible angle, to give you the maximum turning rate. To wax technical for a second, Lift = C.1/2 Rho. V squared. S (C is lift coeff, Rho is air density, V is speed, S is wing area). S is fixed (ignoring things like flaps) so you need max lift, speed and air density. So the max rate of turn is achieved low down at max power and max lift - on the light buffet. Because you're on the buffet the speed is fairly low as there is so much induced drag, but it is the max possible with the power available, not the minimum.
Don't have the gubbins for a min radius turn but I would imagine it's flown at a lower speed, using flaps etc.
Simon 150, normally a 'max rate turn' is accepted as being a max rate level turn. A true max rate turn is flown at the airframe g limit, which in most aircraft will mean descending as they don't have enough power to sustain max g in level flight.