I'm retired and no longer have access to the 737 Minimum Equipment List (MEL). Here's my guess as to how it might have played out to keep the all the papers in order.
Not going back to the gate for a torn seat back pocket, but it has to be deferred in the logbook prior to departure. Pull over, somewhere on the way out, call Dispatch and get a 3-way call with Maintenance.
Write it up in the logbook and fill out an MEL sticker. This would probably be under "Non-Essential Equipment and Furnishings".
Amend the flight release to include the deferred maintenance item. Time and initials please.
Make sure we haven't burned down fuel to less than minimum takeoff for the flight. Yes, the engines have been running all this time. (Think about that when your Honda shuts off at a stop sign.)
Get back in line and figure out how much time we have to make up to not be late according to the D.O.T.
The captain may be the final authority, but you better be able to justify your actions if questioned. Common sense does not always apply.