The electron - described as a four-spinor in the Dirac equation - transforms according to the $(1/2,0)\oplus(0,1/2)$ representation of the Lorentz group, so it is actually a direct sum of a left- and right-handed Weyl spinor. So is the electron technically a set of two other massless fundamental particles?
You are basically asking a circular question of nomenclature. The Dirac quantum field is is a bispinor compactly packaging several degrees of freedom, such as the left- and right-handed Weyl spinors you wrote down the Lorentz transformation properties of. We call both left- and right-handed electrons "the electron", collectively, but of course they are distinct degrees of freedom with different couplings (e.g. electroweak ones) and, naturally, experimental signatures. They are also coupled, so you may think of them as mutating into each other, and so different facets of the same object.
Electrons may have different chiralities, charges (electrons vs positrons), momenta, positions in space, etc... The Dirac quantum field compactly describes all of these states, packaged in its components. So, when you say "the electron", you refer to all of these states, unless it makes sense to be more specific. Don't you do this for all particles? "Technically" does not have the same whiff of quibbling in physics that it has out there...