However the white and bare wires both have the same ground.
Neutral wire bus bar.
The original bar is mounted in a plastic housing which i assume isolates it from the box.
The problem primarily comes from the inappropriately named neutral wire.
The black hot wires are protected from overload by circuit breakers in the panel which typically sense the current in the hot wire and trip out if current is exceeded for too long.
If that voltage is not the same it will trip gfci.
There is also a bare ground wire from the neutral bar to the water pipes.
The white wire is neutral not ground.
The bare wire is a ground to.
Your answer is yes you do put in the bare wire with the white wire.
Some gfci breakers have between 30 80volts on neutral.
A double tapped neutral is when more than one neutral wire is fed into a single screw terminal on the neutral bus bar in the main electric panel.
Neutral bus bar once the power leaves the electrical service panel through the hot wire s of a circuit and does its work through the electrical devices light bulbs outlets etc the electrical current returns back to the service panel through the neutral usually white circuit wire which is connected to the neutral bus bar.
Neutral bars are widely used in residential and commercial electrical service panels to terminate all those white neutral or return wires from the many load circuits in the building.
There is nothing neutral about a neutral wire.
You can see this clearly in the picture below as there are multiple neutral wires feeding into a single screw in more than one instance in this spaghetti mess of wires.
It is a current carrying conductor just like a hot wire and has all the potential for danger and should be treated with the same respect.
Consider the schematic for a flash light.
The main feed is 3 wire with 2 hot legs to the main breaker and an aluminum wire to the neutral.
Gfci s work differently and can put out a voltage on the neutral to be reconciled later on the ground bus.
And on spas if ground wire is loose or water filled equipment you can have that voltage in water.