Does a planeswalker with two loyalty counters survive being dealt two damage from a single source?

When a source deals damage to a planeswalker, that damage is converted into the removal of that many loyalty counters (CR 120.3). So 2 damage removes exactly 2 loyalty counters from the planeswalker, leaving it at 0 loyalty counters.

A planeswalker with 0 loyalty counters is then put into its owner's graveyard as a state-based action (CR 704.5i). This check happens after the damage event fully resolves, not during it. So the planeswalker does survive the damage step itself — it simply dies immediately afterward when state-based actions are checked.

In practical terms, 'surviving' damage and 'surviving the turn' can feel different here: the planeswalker is not destroyed by the damage directly, but it will be put into the graveyard the moment state-based actions are checked, which is before any player receives priority again. It does not remain on the battlefield.

Example: Your opponent controls Jace, the Mind Sculptor with 2 loyalty counters. You cast Lightning Bolt targeting it. The Bolt deals 3 damage, removing 3 counters — but Jace only has 2, so 2 are removed leaving 0. State-based actions are checked, and Jace is put into the graveyard. If instead only 2 damage were dealt (e.g., a 2/2 creature attacking it), the same result occurs: 2 counters removed, 0 remain, Jace dies to state-based actions.

So while the planeswalker technically 'survives' the damage resolution itself, it is immediately placed in the graveyard due to CR 704.5i. It will never be seen with 0 loyalty counters on the battlefield by any player.