Time independent perturbation theory Why do we talk of transitions only in time-dependent perturbation theory, when the eigenstates are corrected even in time
independent perturbation theory?
If we can, for sake of argument, say: eigenstates of the system in TIPT changes (get corrected) and so does state of the system before and after the perturbation, from eigenstate to another of the original  Hamiltonian of the system  in TDPT, then why no transitions in latter.
 A: The difference is in the context = formulation of the problem:
Time independent perturbation theory is a method for calculating approximate eigenvalues and eigenstates of a Hamiltonian, which cannot be diagonalized exactly. There are no transitions, because the Hamiltonian was always the same, and if the system is put in a state, it will remain there (no transitions).
Time dependent perturbation theory is busy with calculating transition rates: the system is prepared in a particular state and literally makes transitions to other states, the rate of which is often expressed by the Fermi golden rule.
Thus, in the former case we are looking for the eigenvalues and eigenstates of a static problem, in the latter for the transition rates of a dynamic one.
The mathematical formalisms overlap: the time dependent perturbation theory becomes time independent in the limit of low frequencies, and one often uses time dependent adiabatic switching for time-independent problems.
A: In my answer attempt, I take a detour of analysing TIPT via TDPT to get the concept of transition.
With great care, I believe one can derive time independent perturbation theory from time dependent perturbation theory by considering a infinitely slowly varying perturbation. Starting at t=$-\infty$ with zero strength and reaching nominal strength at t=0. Infinitely slow perturbation keeps the system constantly in it's eigenstate, which was a ground state in this case. (See Gell-man and Low theorem) Only exception is that the perturbation changes the groundstate for example via un-avoided crossing of an excited state with a ground state (leading to a degenerate ground state). In other words, the system is perturbed adiabatically by keeping it constantly in its ground state. (Great care here excludes pathological cases such as degenerate ground state)
Transitions, on the other hand are defined as processes where system transfers from one eigenstate of the Hamiltonian to another. This requires a changing field as a perturbation (a field with a finite time derivate). But is discussed above, due to adiabatic perturbation, the system is constantly in its groundstate and no transitions occur by definition. To recapitulate, the reason why TIPT does not induce any transitions, is that it can be traced back to TDPT with infinitely slowly varying perturbation and an adiabatic process.
Edit: another way to see it: response function derived from TDPT does not have poles (indicating absorption, i.e. a transition) at zero frequency unless the system has a degenerate ground state. Therefore, no transitions can be induced by infinitely slowly moving field. And if the ground state is degenerate, the adiabatic assumption will fail and that is a different story then.
