Damage Trample and Lifelink which cause Damage
Assuming nothing else happens, in the combat damage step, P1's creature will deal 2 damage to P2 (and 1 damage to P2's creature), and P2's creature will deal 1 damage to P1's creature.
Lifelink modifies the damage dealt, so at the very same time, both creatures are dealt 1 damage, P2 is dealt 2 damage, and P2 gains 1 life.

119.3f. Damage dealt by a source with lifelink causes that source’s controller to gain that much life, in addition to the damage’s other results.

The artifact triggers, but before the trigger is put on the stack, State-Based Actions are checked and see P2 at 0 life, causing that player to lose the game.

116.5. Each time a player would get priority, the game first performs all applicable state-based actions as a single event (see rule 704, “State-Based Actions”), then repeats this process until no state-based actions are performed. Then triggered abilities are put on the stack (see rule 603, “Handling Triggered Abilities”). These steps repeat in order until no further state-based actions are performed and no abilities trigger. Then the player who would have received priority does so.
704.3. Whenever a player would get priority (see rule 116, “Timing and Priority”), the game checks for any of the listed conditions for state-based actions, then performs all applicable state-based actions simultaneously as a single event. (...)
704.5a. If a player has 0 or less life, he or she loses the game.

The trigger from the artifact is never put on the stack, and certainly never resolves.