If the player is controlling a sphere and I want the game to end when the sphere enters a goal volume (a cube collder), how can I detect if the sphere is entirely/95% inside the trigger, not just touching it?
How about getting the distance between the sphere's position and the cube collider's position...if it's a smaller distance, then a larger percentage of the sphere is inside the cube, more or less. That would only work perfectly for two spheres, but it would generally work for a sphere and a cube.
I would set up multiple gameobjects with triggers that are based on the percentage you want. That's how I've done proximity mines and other proximity based solutions. Sounds like a lot easier of a solution too!
If you are making a soccer type goal, I would just place the collider inside the goal so when it touches the collider and triggers it, it would only count as a goal because you placed it at 95% of the diameter of the ball inside the goal.
You can combine transformation matrix of your Box GameObject and it's BoxCollider.size to get OBB. Once you have a OBB, you can combine rotation matrix of your Box with Sphere's radius and Sphere's position to get 6 points. The only thing left to do is check if these points are located within the OBB. If all 6 points are inside, sphere is 100% inside. Quite simple.
A sphere is approximately 95% inside a trigger if it is touching a smaller trigger.
So simply make your trigger smaller.
In this case, make the dimensions of the trigger smaller by [some value] of the diameter of the sphere. You can calculate [some value], or, just use an easy value (like 95% of the diameter) This is, like I said, just an approximation, but you wouldn't be able to tell it visually from the real thing.