I read that other thread about the definition of (minimum 1) and I believe it was concluded that it was referring to the minimum amount of damage the attack can be reduced to.

Which brings me to the question: if the attack is already at negative damage (from Amy's Assistance/Dual Wielding/etc.), would using Cage Arena reset the damage back to 1?

nope. is going from 0 to 1 a reduction? no sir it is not. Just how fight or flight cannot reduce the attack from below printed to printed.

minimum 1 is just how low the ability can bring the damage of the attack. If it's already below that, it just won't affect the damage.