Earlier quoted context omitted.
No. Running a red light is when your possession crosses an intersection while the light is red, and should apply to the registered owner.
If I lend a neighbor my kitchen knife and they murder someone with it, should I be liable?
While a bit more extreme than your example, there have been multiple cases where the parents of a school shooter have been held responsible because they provided access to a weapon when there were warning signs.
On the less extreme end of the spectrum, this is the same reason why you have to pretend that you are buying a "water pipe for tobacco" and not a bong if you don't want to get kicked out of the headshop (in places where that is still illegal).