What does it mean when it says gearbox default light comes on, it when taking off, it's kinda jerky. 2002 s type jag 4.0 had the transmission rebult 2 yrs ago.

 

Reading very carefully what you have written, and a few things come to mind.

How many miles since the rebuild?.

Since the rebuild (2 years) has the fluid ever been changed/serviced?.

The "kinda jerky" is usually clutch/band slippage, and that can be caused by old fluid, low fluid, wrong fluid, electrical gremlin, etc, etc.

Sometimes by disconnected the battery, ley it sit for a few minutes, and reconnecting can sort this, sometimes not.

Maybe a code read of the car will assist in what is doing the wrong thing.

 

The two electronic control modules (ECMs) for the engine and gearbox communicate with each other and a 'trans fault' can sometimes be nothing more than a weak battery or a bad coil pack on one of the spark plugs.

If you can have the codes read, please do so and post them here, in the meantime, have the battery tested to see if it is still serviceable.

Does the vehicle have any other warning lights that come on or flash when accelerating?

 

The 4.0 does not have a separate computer module. It's all done by the PCM.

Could easily be:
failing battery
damp in trunk
an engine fault (probably minor) causing gearbox to react poorly

Investigation needed...