A "bucket" is nothing more than a fare or ticket price. Coach has 6 possible buckets/fares and the sleepers each have 5 possible buckets/fares. Each of these remains a fixed value for some period of time ranging from a few months to as much as a year.
Amtraks computer (Arrow) decides which bucket to offer for sale on any day and can change what's offered from day to day. F'rinstance, here are the current buckets/fares for the whole route of the Empire Builder between Chicago and Seattle...
View attachment 24185
...listed in the order to Coach, Roomette, Family Bedroom and Bedroom. You might find that a Roomette fare is offered at $651 when you do a trial booking today for some future date. Then tomorrow you might find the fare has gone up to $767 for travel on that same future date in a Roomette.
In short,
bucket is nothing more than high-falootin' term for ticket cost or fare - just part of the jargon of this forum. I call it gobbledegook!