Approx how large of an area should be dug out 6-8"s thick for the posts on a 2 post lift?
Don't do this. It will make your floor (and lift) weaker. Best to follow the instructions and pour a flat, consistent slab. If this was a good idea... the manufacturer would give you instruction on how to do it.
For a new slab, flat and consistent is the strongest way to go. Pad pouring is for retrofit applications where you don't have a floor with the required thickness or crack/structural problems. Pads also have pin/key systems to integrate the pads with the larger floor, which you can not easily replicate during a new pour.
Will you be putting rebar in your floor? If so, the different thicknesses will cause you additional issues and can promote cracking. (Exactly the opposite of what you'd want.)
It's easy to assume that having "footers" for the posts is a strong and good thing to do. The vertical force on your slab from the two post lift is not really any greater than just having your car parked directly on the slab. (Your car's weight sits entirely on 4 tire contact patches and the weight of your lift sits on 2 base plates - which are larger and spread the forces out more.)
The forces of greatest concern for your slab hosting your 2 post lift are tipping forces. Having a continuous and consistent slab will give you the best control of any shrinkage cracking that may occur and the strongest slab for your lift.