I'm still looking at where "deep discharge" was originally used by Boeing. Tried browsing that video for an hour, but missed it. Pointers welcome!
Webcast - Boeing 787 Updates
"Deep discharge" of a cell, without context, is a well-known phenomenon in all kind of batteries, where open-circuit cell voltage drops below some rated minimum. At least in some cells technologies (I do not know for Lithium Cobalt Oxide, much less the particular type used), deep discharge can permanently damage a cell, increasing its internal resistance. If that occurs, it is possible that the cell later overheats when subjected to high current, e.g. from a charger, or from other cells powering the load; the later can only happen in batteries with more than say 3 or 4 cells in series, and with one cell that developed significant imbalance w.r.t. the other ones.
A short circuit inside a cell (even a mild one, usually called leakage, as could occur from moisture) can
cause deep discharge of a cell, but is not by itself deep discharge.