What the board does with your tasks
This page keeps no list of its own. It reads the same task store the task manager writes, and shows everything in it except the Completed and Reference buckets, so anything captured in the task manager an hour ago is already in the tray. The only thing stored here is the sort itself: a task identifier against a quadrant name, under cc-eisenhower. Sorting is an overlay, and nothing about the task changes when you drop it into a box. Adding a task in the field above goes the other way, writing it straight into Next actions.
A worked pass
Say eighteen open tasks arrive in the tray. You put three in Do first, six in Schedule, four in Delegate and two in Drop. The four counters across the top then read 3, 6, 4 and 2, and the Unsorted header reports three still waiting, because fifteen of the eighteen have been placed. Those counters only describe open work, so completing a task elsewhere quietly takes it out of its quadrant and off the counter.
Which controls touch the real task
Three of them do. The date field on a Schedule card writes a due date onto the task and moves it into the Scheduled bucket, and emptying that field sends it back to Next actions. Move to Someday, on a Drop card, changes the bucket and leaves the task alone otherwise. Delete removes it from the shared list for every tool on the site, which is the only destructive button here. The dropdown and the small cross do not: one moves a card between boxes, the other returns it to the unsorted tray with the task untouched.
A task deleted somewhere else leaves its old quadrant entry behind in storage, and the board ignores an entry whose task has gone rather than tidying it away. Opening this page never rewrites your storage on its own, which is what you want from a tool sharing one list with the task manager, the daily planner and the timer.