I think your right about it being amaranth and I'm leaning
heavily towards it being "redroot pigweed" (Amaranthus
retroflexus L.).
http://www.wes.army.mil/el/pmis/plan.../amarant1.html


A .retroflexus looks like a good candidate. There is also A. palmeri which is
similar.
According to the book Weeds of the West, the two species will hybridize.
I haven't eaten amaranth, but the seeds are used as a grain and a flour I
believe, I had a good article on amaranth, but I may have thrown it out in one
of my cleaning frenzies. Will look for it.
There is an Anagallis, but the weedy one that I know of is Anagallis arvensis,
the Scarlet PImpernel, and is a tiny, prostrate plant. Anagallis and Amaranth
are not even in the same Family.
You asked what the flowers might be called: flower spike, or inflorescence,
or flower clusters all could describe them.
You wondered how they might have gotten the probably birds that ate the
seeds.

Emilie
NorCal