LS attacks, declares his DS objective. After the edge battle, DS attempts to play Backstabber to turn a 1-on-1 attack into a 1-on-2. However, Backstabber's reaction says to deal 1 damage to the engaged objective after this card enters play during an engagement... which it has just now done. And the engaged objective is his own... and it's 1 point away from being destroyed.

Has Backstabber just destroyed a DS objective? Or is the reaction designed to damage/destroy a LS objective at DS's discretion?

All triggered abilities (Actions, Reactions, and Interrupts) are optional unless they say "forced" (See Rancor for example).

Backstabber is most effective being flashed in as an attacker to get an extra damage on a LS objective, but it can be used defensively and you can choose to not use the Reaction and thus not damage your own objective.