Do you size the grounding conductor from the breaker or the wire?
From the breaker. NEC 250.122 sets the equipment grounding conductor size by the ampere rating of the overcurrent device ahead of the equipment, not by the circuit conductors running with it. One rule caps that: per NEC 250.122(A), the conductor is not required to be larger than the circuit conductors it serves.
Why does the overcurrent device set the size?
Because the equipment grounding conductor is not sized for load. It is sized for the fault. Its job is to carry fault current back to the source long enough and hard enough for the breaker or fuse to open, and the thing that describes that event is the rating of the device that has to clear it. So the table is indexed to the device.
The circuit conductors are answering a different question. They are sized for the current the load draws under normal conditions. That is a separate calculation with separate inputs, which is exactly why reading the grounding conductor off the phase conductors is the wrong move even though it feels intuitive.
What does that look like on the page?
You find the device rating, you pick a material, you read the row. At a 100 A overcurrent device the minimum is 8 AWG copper or 6 AWG aluminum. At 200 A it is 6 AWG copper or 4 AWG aluminum. At 400 A it is 3 AWG copper or 1 AWG aluminum. Those all come straight out of NEC Table 250.122, which is the whole method: the table already did the thinking.
The Cable Tray Bonding Jumper Size Calculator is that lookup with the page-turning removed. Pick the device rating, pick copper or aluminum, get the minimum size and the article it came from. For devices above 2000 A, go to Table 250.122 directly. It is one of the tools in the catalog, each of which exists to answer a single question and then get out of the way.
What is the cap in 250.122(A)?
The bonding jumper or grounding conductor is not required to be larger than the circuit conductors it serves. That is the one place the wire does enter the answer, and it only ever works one direction. It can hold the size down. It never drives it up.
Treat the table value as a floor and that cap as the exception you check, not the rule you start from. If someone hands you a grounding conductor smaller than the table for that device and cannot point at 250.122(A), the number is wrong.
Where does the breaker rule stop applying?
On the line side. A supply-side bonding jumper, meaning one on the line side of the service or of a separately derived system, is sized from the supply conductors using NEC Table 250.102(C)(1), not from an overcurrent device. Different table, different input, same-sounding name. That mix-up is common enough to be worth saying out loud: 250.122 covers load-side equipment bonding jumpers and equipment grounding conductors, and nothing else.
How does this apply to cable tray?
Identically. Find the overcurrent device protecting the circuits in the tray, then read Table 250.122 for that rating and your conductor material. Nothing about the tray changes the sizing method for the conductor.
What the tray can change is whether you need a separate conductor at all, which is a different section of the code and is worked through in whether cable tray can be used as the equipment grounding conductor. If you are choosing between materials at the same device rating, the two columns are compared in copper versus aluminum grounding conductor size.
Values follow NEC Table 250.122. Codes are adopted and amended locally and change between editions. Confirm against the NEC edition your jurisdiction enforces and the project specifications before you build. This is a reference, not a stamped engineering decision.