quantum field theory - Zero modes ~ zero eigenvalue modes ~ zero energy modes?

There have been several Phys.SE questions on the topic of zero modes. Such as, e.g.,

Here I would like to understand further whether "Zero Modes" may have physically different interpretations and what their consequences are, or how these issues really are the same, related or different. There at least 3 relevant issues I can come up with:

(1) Zero eigenvalue modes

By definition, Zero Modes means zero eigenvalue modes, which are modes $\Psi_j$ with zero eigenvalue for some operator $O$. Say,

$$O \Psi_j = \lambda_j \Psi_j,$$ with some $\lambda_a=0$ for some $a$.

This can be Dirac operator of some fermion fields, such as $$(i\gamma^\mu D^\mu(A,\phi)-m)\Psi_j = \lambda_j \Psi_j$$ here there may be nontrivial gauge profile $A$ and soliton profile $\phi$ in spacetime. If zero mode exists then with $\lambda_a=0$ for some $a$. In this case, however, as far as I understand, the energy of the zero modes may not be zero. This zero mode contributes nontrivially to the path integral as $$\int [D\Psi][D\bar{\Psi}] e^{iS[\Psi]}=\int [D\Psi][D\bar{\Psi}] e^{i\bar{\Psi}(i\gamma^\mu D^\mu(A,\phi)-m)\Psi } =\det(i\gamma^\mu D^\mu(A,\phi)-m)=\prod_j \lambda_j$$ In this case, if there exists $\lambda_a=0$, then we need to be very careful about the possible long range correlation of $\Psi_a$, seen from the path integral partition function (any comments at this point?).

(2) Zero energy modes

If said the operator $O$ is precisely the hamiltonian $H$, i.e. the $\lambda_j$ become energy eigenvalues, then the zero modes becomes zero energy modes: $$ H \Psi_j= \lambda_j \Psi_j $$ if there exists some $\lambda_a=0$.

(3) Zero modes $\phi_0$ and conjugate momentum winding modes $P_{\phi}$

In the chiral boson theory or heterotic string theory, the bosonic field $\Phi(x)$ $$ \Phi(x) ={\phi_{0}}+ P_{\phi} \frac{2\pi}{L}x+i \sum_{n\neq 0} \frac{1}{n} \alpha_{n} e^{-in x \frac{2\pi}{L}} $$ contains zero mode $\phi_0$.

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Thus: Are the issues (1),(2) and (3) the same, related or different physical issues? If they are the same, why there are the same? If they're different, how they are different?

I also like to know when people consider various context, which issues they are really dealing with: such as the Jackiw-Rebbi model, the Jackiw-Rossi model and Goldstone-Wilczek current computing induced quantum number under soliton profile, Majorana zero energy modes, such as the Fu-Kane model (arXiv:0707.1692), Ivanov half-quantum vortices in p-wave superconductors (arXiv:cond-mat/0005069), or the issue with fermion zero modes under QCD instanton as discussed in Sidney Coleman's book ``Aspects of symmetry''.

ps. since this question may be a bit too broad, it is totally welcomed that anyone attempts to firstly answer the question partly and add more thoughts later.