Fermi's guess at the Beta Decay Hamiltonian

First, your question has a history of science part on the 1933-34 Fermi paper, here translated in English (Wilson, F L (1968), "Fermi's Theory of Beta Decay", American Journal of Physics 36 (12) 1150–1160). This is the first nontrivial application of QFT in solving an actual problem, so an epochal linchpin paper of the 20th century. Fermi was flying blind on this: he created the annihilation/transmutation field, almost single handedly. He did not guess the complete operator resolution your (1.1) attributes to him--this is what he would have done had he been sensitized to the chirality and parity structure of the weak interactions, thinking that became mainstream only in the mid 1950s, surveyed by many experiments.

He naturally grouped "heavy particles", nucleons, together, as he had already accepted Heisenberg's 2-state (isospin) nature of the nucleon, and (boldy!) adopted Pauli's wispy neutrino hypothesis, regarding the neutrino as a similar, "light" partner (lepton) of the electron. So, he naturally grouped heavy particles together, and light particles together, and mused on spin-orbit interactions.

However, I suspect your real question is an attempt to understand the Fierz transposition, which you text should be covering. If it does not, toss it with extreme prejudice, and master Okun's delightful and trenchant "Leptons and Quarks" first, which, in my strong opinion, you should have done in the first place--one assumes you are trying to learn physics, after all.

The point is the dancing couples of Fermions with their spinor indices saturated may change partners and saturate with their indices, by dint of combinatoric properties of the Clifford algebra tensor products, like, $$ bar{chi}_1 gamma^mu (1+gamma_5)psi_2 ~~bar{psi}_3 gamma_mu (1-gamma_5) chi_4 = -2 bar{chi}_1 (1-gamma_5) chi_4 ~~ bar{psi}_3 (1+gamma_5) psi_2 . $$ Fermi simply assumed Lorentz invariance, which, in this language, amounts to saturation of all Lorentz indices, but he never considered parity, so he did not throw in a set of all $gamma_5 $ combinations to determine from experiments; that was a quarter century before its time. He could have considered and Fierzed cross bilinears, but he had a preternatural ability to dismiss the inessential, and to focus on key points, leaving side issues for others.