Deleting from within the Inbox vs. a Label

Within a special folder like the Inbox, deleting moves the messages to Gmail/Trash, causing it to be removed from ALL other labels/folders.

Within a label/folder, deleting removes the message from the current label/folder only, and the message remains in other labels/folders.  To make deleting remove all labels/folders, go to Preferences / Advanced / General / Config Editor and search for "pb.gmail.label_delete.useAllMail". You'll want to double-click this preferences to change it to False.  

Postbox Delete Preferences for Gmail Accounts

Within Options|Preferences / Accounts / [your Gmail account] / Account Settings, here is what each of the delete settings will do:

Move it to this folder [Trash] 
This is the default setting. In Gmail accounts, you can not specify a different folder for Trash. When using the default settings in both Postbox and Gmail:

If you delete a message from Postbox's Inbox, it is then:

• Removed from the Postbox Inbox, but NOT the Gmail Inbox until you empty the Trash in Postbox.
• Placed into the Postbox Trash folder, AND the Gmail Trash folder.
• Removed from Postbox labels, AND Gmail's labels (although the Inbox will show messages with labels, but label selection will not)
• Removed from Postbox's All Mail folder, AND Gmail's All Mail folder.

Remove it immediately
If you select this setting, and then delete a message in Postbox, it is then:

• Removed from the Postbox Inbox, and the Gmail Inbox.
• Placed into the Postbox Archive folder, and the Gmail All Mail folder.
• A copy is not placed in the Postbox Trash folder.

Just mark it as deleted
The IMAP protocol allows messages to be marked for deletion, a sort of limbo state where a message is still present in the folder but slated to be deleted the next time the folder is expunged. With Gmail's default settings, when you mark a message as deleted, Gmail doesn't let it linger in that state -- it deletes (or auto-expunges) it from the folder right away. If you want the two-stage delete process:

• Select "Auto-Expunge off" under the "Forwarding and POP/IMAP" tab in Gmail Settings.
• Check the "Clean up ('Expunge') Inbox on Exit checkbox in the same Postbox preferences panel listed above.

When you delete a message using the default "Archive the message" options in Gmail settings, it is then:

• Marked as deleted in the Inbox, until you exit Postbox, upon which time, the message will then be removed from the Postbox Inbox, and the Gmail Inbox.
• Placed into the Postbox Archive folder, and the Gmail All Mail folder.
• A copy is not placed in the Postbox Trash folder.