The one mechanism behind all of it
A stone does not crack because it got too hot. Every cordierite stone here publishes a limit of 1450 F and no domestic oven or covered grill approaches it.
It cracks because one part of it got hotter than another part, quickly. Hot material expands; the cold material next to it does not; the resulting tension pulls the body apart, because ceramics are strong under compression and weak under tension. Cordierite is the material of choice precisely because its published expansion coefficient is very low - 2.9 to 4.8 micrometers per meter per kelvin - so a given temperature difference produces less stress than it would in ordinary clay. Low is not zero.
Gradient, not temperature
Every rule about handling a stone follows from this one sentence
If a change would create a big temperature difference across the body quickly, it is dangerous. If it would not, it is fine.
The six causes, most common first
1. Cold stone into a hot oven. The face nearest the element takes radiant heat immediately while the rest of the body is still at room temperature. Always start the stone in a cold oven and bring both up together.
2. Hot stone onto a cold surface. Setting a 500 F stone on a stone or tile counter pulls heat out of one face fast. Let it cool in the oven, or move it to a wire rack where air, not a cold solid, takes the heat.
3. Something frozen dropped onto a hot stone. A frozen pizza straight from the freezer creates a very cold patch on a very hot surface. Let frozen food sit ten minutes first, or bake it on a screen.
4. Water on a hot stone. Same as above but faster, and the steam adds an internal pressure component. This includes the perfectly reasonable-seeming instinct to run a hot stone under the tap.
5. Direct flame or a burner underneath. On a grill, a burner right under part of a stone puts several hundred degrees of difference across a few inches of body. Indirect heat only - full setup in the grill roundup.
6. Moisture already inside the body. The reason soaking and washing a stone are bad advice: absorbed water becomes steam inside the pores on the next heat, and it expands roughly a thousandfold.
Two things that are not causes
Oven temperature. A domestic oven maxes out around 550 F. The stone is rated to 1450.
Leaving it in permanently. A stone that lives in the oven experiences fewer thermal cycles and gentler ones than a stone that is moved for every bake. If it is heavy, leaving it in is the safest thing you can do, not the riskiest.
Can you keep using a cracked stone?
Usually, yes. A stone with a clean crack that still sits flat continues to conduct heat into the base perfectly well - the crack is a gap of a fraction of a millimeter in a plate that is 15 millimeters thick. Two conditions: the pieces have to sit level, and you have to accept that the crack will grow.
Replace it when it has broken into pieces that rock, when a piece is small enough to fall through a rack, or when the crack is wide enough for cheese to run into and burn on the element below.
Do not try to glue it. There is no domestic adhesive rated for repeated 500 F cycles in contact with food.
The permanent fix
If you have broken two stones, buy something that cannot break. Carbon steel and cast iron have no thermal shock failure mode at all - they conduct heat quickly enough that internal gradients stay small, and they yield rather than fracture. The trade is weight and seasoning, and it is covered in stone versus steel.