I can affirm, after doing it today on my Mage, that you do not need to actually be on the Shadow Wardens part of the dailies.
You must have completed:
The Shadow Wardens
HOWEVER:
If you have completed both The Shadow Wardens and Druids of the Talon, and on one day you so happen to choose Into the Fire instead of The Forlorn Spire, you can still "loot" this.
Today, I chose Into the Fire to do my Molten Flow quests, and decided to head over to Widow's Clutch and kill a [npc=52122]. It dropped the [item=69675] on the ground and I was able to pick up the acorn, start The Mysterious Seed, turn in the quest for the 5 marks, AND do the daily that the lasher offered. It seems as though the Daily offered by Anren after doing Need... Water... Badly... is only available when you do Into the Fire, but the daily from the lasher is available no matter which daily you choose.
Hope this helps!