The USGA Handicap Manual creates an interesting nuance between "Holes Not Played or Not Played Under the Principles of The Rules of Golf" and "Unfinished Holes and Conceded Strokes". It would appear from the examples that "Unfinished Holes and Conceded Strokes" is mainly intended for match play. All the examples deal with putting and match play scenarios. Of course, if one is lying 5 in the bunker and your opponent is lying 2 ten feet from the hole, one can still use this particular handicap rule.
"Holes Not Played or Not Played Under the Principles of the Rules of Golf" appears to apply to our unexpected "lost ball" scenario or any other situation where a score for a hole is not achieved through play under the Rules.
One does not have a choice between the two handicap rules. If a player pumps two O.B. and announces "I am done with this hole", that is not considered an unfinished hole. He records par plus whatever handicap strokes he gets along with an "x". In a sense, the handicap system "penalizes" a player who does not play a hole or plays a hole without following the Rules. Now one might think that writing a lower score than one likely would have recorded is hardly a penalty. Except consider that the player who routinely quits a hole or bends/breaks the Rules is going to record a lower score for handicap purposes. So he is going to have a lower handicap and he will be less competitive against players who play by the Rules and complete every hole. Yes, the guy who picks up mid-hole is going to have a vanity handicap and he is going to be ripe for the picking.
In practice, it's worth mentioning that there's absolutely nothing wrong with picking up, moving on, and recording your ESC maximum once you get to that point, even though you didn't "finish" the hole.
As a reminder to all, the discussion we're having here relates to posting of a score for hcp purposes only. It has nothing to do with someone's actual score as it relates to the round itself....