Overheating car over heats when driving

May be heater core or the radiator has a pin hole in it..

The piston can only be damaged during the overheating condition. It is possible, and makes sense, that the damage to the piston was discovered during the test drive.

with heated? Signal , what signal?
what is that?
and if the engine is overheated , that is normal the jerking.
fix overheated cause.

Possible, with rad cap off, let vehicle idle until coolant fan comes on then shut vehicle off for 30 minute cool down. Top up rad, install cap and drive and recheck heater operation and any overheating.

Assuming the radiator fan works correctly, my guess would be a leaking head gasket.

see if the cooling fan for the radiator is working could be a relay or fuse for cooling fan

It sounds like water pump may be your problem. It circulates a 50/50 mixture of glycol (or some form of) and water through the engine block and cylinder head and back through the radiator. In the radiator the mixture is cooled when air passes over the fins of the radiator. This is accomplished by driving or a mechanical/electrical fan when stationary. The water pump also circulates the mixture through the heater core giving the car heat. If the water pump is not circulating the water back to the radiator to cool, it will overheat. More heat is caused by the engine when driving (much less heat when idling), hence the overheating while driving.
I hope this helps and good luck!

Your car have a broken evaporator, you have to remove the dash in order to change it.

Most likely not. Pulling the thermostat will get you home, but not a great fix. Is it overheating driving down the road or sitting still? If it is overheating driving down the road, look for a clogged radiator or stuck thermostat. If it overheats sitting still, look for your cooling fans not coming on.

Generally when the heater isn't heating anything and putting out cold air the heater core is bad. I have never worked on a BMW before and have never heard of a heater core causeing overheating, but because they are linked, it maybe a clog in the core causing poor coolant circulation.