"bucket" is just a name. think of each train as having "buckets" in which available reservations are kept. Each represents a different price level onboard. As one is taken, the "bucket" empties.
Here is a detailed listing of AMTRAK codes.
AMTRAK INVENTORY CODES
Here is a help checklist, including a screen shot
SABRE HELP
So, via their reservation system (and a process called revenue management), AMTRAK can adjust just how many tickets they will sell at which level and for how much.
Airlines do the exact same thing, except they have dozens of "buckets". Next time you buy an airline ticket, look at the first letter in the fare code - that is the "bucket".
Airlines use letter ranging from A to Z, and it is a mess to figure out, what with all the advance purchase restrictions, refundable v non, day of week, over Saturday stay, round trip v one way and on-and-on. But these codes are availalbe to check via online resources so one could easily research availability.
AMTRAK is no where near that complicated, but you can not easily decipher which fare level you are getting. (I suggest that should be correct, btw)
Next time you book, use the chart above and ask the reservation agent at what level the fare quoted is, and for that matter, what the other fares are.