Complete the following Survival Ring challenges without taking any damage.
Remember that if you want your achievement, DONT TAKE ANY DMG! Even campfire near quest giver will fail your quest. Any damage will fail.
So, since you've taken the quest avoid any damage before turn-in
This is correct for both criterias.
And if you do take damage (this includes absorbed damage), you must abandon the quest and pick it back up from the quest giver.