Double strike issue?

stikxs

Was playing a ToA match earlier where I had a defending creature of some type out and the opponent had a creature with double strike and used arlinn's first ability to give trample/berserk/haste. When the creature attacked it hit my blocker first, killed it, the trample damage came over, and then it hit me with the second strike. I'm pretty sure I read here somewhere that this how it is supposed to work.

Fast forward to another match where I have a double striker and the opponent had a creature with vigilance. My guy hit the blocker, killed them, but never had it's second strike (the celebrant gained double strike from it's exert ability, the arborback gained vigilance from a card earlier):

Szamsziel

Without trample double strike damage do not pass to planeswalker.  Your combat cerebrant was blocked. Even if it's blocked with 1/1 - no damage will be done to planeswalker. (chump block): article
http://magic.wizards.com/en/articles/archive/limited-information/chump-blocking-2012-08-22

stikxs

So the 2nd strike won't happen when the first kills the defender and trample is not present?