Not all limited slip differentials are the same. Even if your truck and your Corvette are both plate-type LSDs, they will likely behave very differently based on the applications. The Corvette is going to be way more aggressive than the truck.
There are three tunable pieces to a plate LSD: Break-away torque (preload), the ramp angle, and the total locking force.
Break-away torque is the measure of required torque to initially allow spinning from side to side. If you were to jack up the vehicle, hold one tire stationary and attempt to rotate the other side, the point at which rotation happens is the break-away torque. This is primarily for low speed turning. If the break-away torque is too high, the rear tires will chirp and squeak in parking lot-type maneuvers as they lose traction before the break-away torque is reached. Bigger tires, heavier vehicles, and stickier tires can get away with more break-away torque.
Ramp angle is how aggressive the LSD action is. As torque input increases (more power!) and/or wheel speed from side-to-side increases (wheel slip), the mechanism that applies pressure to the clutches rotates. Think of this mechanism like an eccentric cam: The more it's rotated, the tighter it gets. The eccentric can be a different shape, more or less angle, and that determines how aggressive that pressure is applied.
Total locking force is exactly what it sounds like: How much total force is applied to the clutches.
The LSD in your truck is almost guaranteed to be less aggressive than one in a Corvette in every way: Less preload, less aggressive ramp, and less overall locking force. I assume you weren't making a conscious effort to tear up your gravel drive, so you probably just didn't have enough input torque or wheel speed differential to really ramp in the LSD. You probably weren't tearing up the driveway with the Corvette either but it likely just has a higher preload that couldn't be overcome on a loose driveway.