I think the choice of the terms themselves really is just historical baggage. Fine structure corrections to the atomic spectrum can only be observed at relatively high definitions, and hyperfine corrections require even higher definitions.
However, in modern usage the terms have significantly specialized. The fine structure correction to the (hydrogen-like) atomic Hamiltonian are the first order terms in the non-relativistic expansion of the Dirac equation. Even though they are often introduced as the relativistic correction to kinetic energy, the spin-orbit coupling, and the Darwin term, they really have a unified origin, the Dirac equation.
The hyperfine corrections result from the interaction of the nuclear multipole moments (excluding the electric monopole) with internally generated fields. The effects of nuclear spin are sometimes introduced as the next step of spin-related corrections after the spin-orbit term, but a more unified treatment would probably group them together with electric quadrupole effects and so on.
With these definitions, it is easy to see why Lamb shift is not included in either fine structure or hyperfine structure, as it results from vacuum fluctuation and cannot be described by the Dirac equation, or multipole interactions.