This is a very good suggestion, and of course good pictures. This idea is highly feasable, and if JageX sees this, I think there would be a high chance they would add this into the game.
There are a few problems though. Sometimes people like to organize their banks into lines, like put all their PKing gear at the top and training gear on the bottom. Could there possibly be a way this could be implemented into your idea so people can make combinations?
yes, their would be an auto / manual deposit button (kind of like the note / item buttons) which would make it so it either manually deposited the item into the tab u are looking at currently, or automaticly, which would automaticlly put it into the right tab.