Is there a standard convention for whether the term "handedness" refers to helicity or chirality? I was under the impression that the "handedness" of a massive spin-1/2 particle refers to its chirality rather than its helicity. This answer, this one and Srednicki's QFT textbook seem to use the term in that way. But this answer and the web pages here, here, here, and here use the term "handedness" to mean helicity.
I often see the statement "only left-handed neutrinos interact (non-gravitationally) with the other Standard Model particles", but (incorporating neutrino masses) this statement clearly only makes sense if "handedness" refers to chirality, because whether or not two particle species interact is clearly Lorentz invariant, like chirality but unlike helicity.
Is the usage of the term "-handed" consistent enough that we can identify one of these sets of sources as "wrong"? Or is the usage inconsistent enough that the word is fundamentally ambiguous and should never be used in technical discussions involving massive spin-1/2 particles without specifying its definition?
 A: The uses are absolutely inconsistent. The problem is that "handedness" is a wonderfully intuitive term for both "chirality" and "helicity". That means that courses and textbooks will use "handedness" to describe whichever word is used more. There is no standard.
Particle physics books that avoid QFT have no need for chirality, so they use handedness to refer to helicity. For example, Griffiths' standard book states on p.138

Neutrinos are left-handed; antineutrinos are right-handed.

in all caps for emphasis. He means there's a left-chiral neutrino field that makes left-helicity neutrino particles and right-helicity antineutrino particles, but he has no need for fields at all. 
On the other hand, books that focus on QFT will likely use handedness to refer to chirality, and this is doubly true if the book spends a lot of time on classical field theory. Less careful books such as Zee's will freely switch back and forth between the two definitions. Personally I try to completely avoid using the word "handedness" at all.
Pop science typically focuses on particles because they're less abstract, so they'll use handedness to refer to helicity. However, you should take caution trying to fit popsci statements you "often see" into a cohesive framework, because half of these statements have been known to be wrong for decades. Almost every popular source still believes that virtual particles can "borrow energy from the vacuum" by the uncertainty principle, that the mass of a particle increases with its speed, that the Higgs gives mass to the proton, that antimatter falls up, and so on. The fact is that there are two Standard Models, a real one made of mathematics and a fake one patched together from dubious analogies, and in the public sphere the fake one has completely won.
