Helicity: projection of spin onto motion. Since neutrinos are massive, I can always move to a reference frame where their motion is towards the opposite direction, meaning I should reverse their helicity. Yet it's said "neutrinos can only be left-handed"; so how do you deal with this change of reference frames? Is the quote actually wrong?
Yep. You're right. Since neutrinos have mass, they cannot be purely left handed if they are the same kind of fermion as everything else in the standard model. This idea - that neutrinos were purely left handed, came from before the neutrinos were known to have mass. In fact, it worked out quite nicely back then. Neutrinos were only made as left-handed particles in weak interactions, and if you believed they could only exist as left-handed particles, there was no way to write a mass term for them in the standard model. All good - they don't have mass because they're only left handed, and the weak interaction only couples left handed neutrinos to left handed electrons (only the left-handed Weyl spinors transform under SU(2) in the standard model) because right handed neutrinos don't exist.
But then neutrino masses were discovered in the form of neutino mixing :(. We could easily give neutrinos a mass by begrudgingly allowing them to have a right handed spinor and giving them the same kind of mass term every other particle has in the standard model. But that seems a bit unsatisfying - then there's no underlying reason for the small neutrino masses - they just happen to have a much smaller coupling than other standard model particles. So people have introduced fancier theories, like the seesaw mechanism and majorana neutrinos which allow neutrinos to have mass, and explain why that mass is small. This model makes neutrinos a fundamentally different kind of fermion to all the other ones in the standard model. This is why I think neutrinos are the #1 particle to look out for for the next step in our understanding of the universe, and why experiments like DUNE, KATRIN, and neutrinoless double beta decay experiments meant to improve our understanding of neutrino masses are so important.
The helicity of a particle is conserved, but is not an invariant under a Lorentz transformation, i.e., a change of reference frame.
The chirality, on the other hand, is an Lorentz invariant and does not change under a reference frame change.
It just so happens that for massless particles, helicity and chirality are coincidental. This also nicely shows that neutrinos ought to have rest mass, although it is indeed very small, since we have only seen left-handed neutrinos and right-handed antineutrinos.
I recommend the Wikipedia article on helicity for more details.