Just for reference I will paste in my argument into this thread also:
I took the definition from the rulebook (they define units as shown below, where organic is the default):

Organic units are all units not robotic or inorganic, as defined in the rulebook. A lot of the psysic powers will not affect inorganic units, but will by default affect anything not inorganic. Such powers will affect falcons and this also shows that e.g. falcons are organic and therefore living? (impossible to be organic and not living)
Therefore falcons, fireprism etc. are organic according to the definition of the rules as written?
As I see it there are 4 categories and each unit is one of the 4 (angelic does not exsist, so rather 3) and organic units are the living units otherwise I see no reason to define the categories in the rules if they are not used for defining living units.
Thats just my understanding of the rules, but I can see where the confusion about living eldar comes from. IMO it should say organic eldar to follow the wording put forth in the core rulebook. No need to have different definitions in each book, but use the "central" definition of units.
P.S. I fully argree that the avatar, wraith constructs etc. are not "living" eldar units.