right, the answer is "it depends"
as is the answer to so many interesting questions..

if the battery is good but just needs to be charged, a thin wire carrying a handful of Amps for a few minutes will recharge it enough to start the car.
The actual starting current comes from the recharged battery, not via the cables.

but if the battery or connections are actually defective, and the current to actually start the car needs to come directly via the cables, heavy wire will be needed.

m