It depends on where you are. In the US, you or your insurance pays first; then you collect from the person who hit you for the cost of the hospital bill, damages, and pain/suffering. Get a good lawyer to haggle with the insurance of the other person.
Then the driver would assume all liability for the accident - including monetary damages.
The car that was moving when they struck each other...
Yours
The owner of the car that caused your damages will be responsible to pay damages to you unless you live in a no-fault state. In that case, your insurance pays for your damages.
that person who hit you
She does if your claim is corroborated by a GOOD witness. (One on your side) It depends: If you were in your neighbors driveway without their permission you were trespassing and you assume responsibility for any damages. However, If you were there with the neighbors permission and the daughter is 18+ then she is responsible for damages.
the car that that did the original striking will pay for all damages.
Typically that is the way it works, They car behind who hit your car pays for your damages and the car in front that you hit will be covered by your insurance.
you would be the payer of the damage done to your own car.
I'm pretty sure the first car that hits and starts the accident pays.
They do ! THEY collided with YOUR gate. It was up to them to pay attention to what was happening around them.