"Chandra, Heart of Fire" damage and Chandra's Incinerator

When you use Chandra, Heart of Fire's first ability to do 2 damage to opponent's planeswalker, Chandra's Incinerator doesn't treat it as non-combat damage.  It doesn't do 2 damage to a target creature, and it doesn't reduce the cost by 2 if it's in your hand.  

Chandra's Pyreling does get the +2/+0 and double strike in this situation, though.