Haha, yes it's confusing.
Let me simplify.
The "depot" as it's called for some reason, where you go to "donate" tritium to the carrier, is your main fuel tank. It can hold 1000T, and is used when jumping.
Your cargo hold is what you see from the right-panel of your ship, go to Inventory, then Transfer (at the top).
You can also see your cargo hold from the Commodities interface.
So to refuel, you have to get Tritium into your ship, then "donate" it to the depot. Other CMDRs can also donate, so maybe that's why they called it that.
To get Tritium in and out of your ship, and into and out of your carrier, you use the Transfer button in the Inventory panel of your ship, as above.
It's a convoluted system that makes little sense, but it gets the job done once you understand how it works.