gauge theory - Axial anomaly in QCD VS axial anomaly in current algebra QCD

I would like to understand the distinction between an axial anomaly in QCD (Theta Vacuum: axion -> 2 gluons) and an axial anomaly in QCD of current (Chern–Simons term: pion->two photons, photon->three pions, ...). A more specific question: is the current axial anomaly related to the topological properties of the theory like "internal" axial anomaly?

If you don't understand question there are clarifications about "internal" axial anomaly and current axial anomaly (as I see it):

1) First, in quantum chromodynamics, a violation of the axial group $U_{A}(1)$ leads to a nonconservation of the axial current:

$$\begin{gather}\label{nonconservation of curent} \partial^{\mu}J_{5,\mu}=2\,i \, \bar q \, \hat m_{q} \gamma_{5} \, q+\frac{N_{f}\,g^{2}}{8\pi^{2}}\epsilon_{\mu\nu\alpha\beta}\,tr(G_{\mu\nu}G_{\alpha\beta}) \;\ , \end{gather}$$

where $G_{\mu\nu}$ - gluon field strength tensor. The violation of the axial group is connected with the fact that the vacuum of quantum chromodynamics has a complex topological structure, and this eventually leads to an additional term in the Lagrangian:

$$\begin{gather}\label{theta term} \mathcal{L}_{\theta}=\theta\frac{g^{2}}{16 \pi^{2}}\epsilon_{\mu\nu\alpha\beta}\,tr(G_{\mu\nu}G_{\alpha\beta}) \end{gather}$$

2) Second, in addition to the "internal", anomaly of chromodynamics written above, there are external anomalies in the chromodynamics of external currents, the simplest of which corresponds to the process $\pi_{0}\rightarrow\gamma\gamma$:

$$\begin{gather}\label{nonconservation of curent in algebra curents} \partial^{\mu}J^{em}_{5,\mu}=2m(\bar q \, \gamma_{5} \, \tau_{3}\, q)+\frac{e^{2}}{16\pi^{2}}\epsilon_{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta} \;\ , \end{gather}$$ where $q$ - quark field, $F_{\mu\nu}$ - electromagnetic field strength. The corresponding Lagrangian for the anomaly has the form ($\bar q \, \gamma_{5} \, \tau_{3}\, q=f_{\pi}m^{2}\pi_{0}$): $$\begin{gather} \mathcal{L}_{em}=-\frac{N_{c}\,e^{2}}{96 \pi^{2}f_{\pi}}\epsilon_{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}\,\pi_{0}\, \end{gather}$$

I think this violation is not related to the topological properties of the theory.

In addition to this anomaly, there is a huge number of others, for example an anomaly corresponding to the process $\gamma\rightarrow\pi\pi\pi$. In order to describe all the anomalies, the Wess-Zumino-Witten action is used. This is possible due to the following statement: any non-Abelian anomaly in four-dimensionality can be represented through the action of Wess-Zumino-Witten in five-dimension (Chern-Simons term) (for further information please refer to Can the effective vertex for $\gamma\to3\pi$ be derived directly from the anomaly?, Chiral anomaly in odd spacetime dimensions).

$$\begin{align} W &=-\frac{iN_{c}}{96\pi^{2}}\int^{1}_{0}dx_{5}\int d^{4}x \epsilon^{\,\mu\nu\sigma\lambda\rho}\,Tr \Bigl[-j^{-}_{\mu}F^{\mathcal{L}}_{\nu\sigma}F^{\mathcal{L}}_{\lambda\rho}-j^{+}_{\mu}F^{\mathcal{R}}_{\nu\sigma}F^{\mathcal{R}}_{\lambda\rho} \nonumber \\ &-\frac{1}{2}\,j^{+}_{\mu}F^{\mathcal{L}}_{\nu\sigma}\,U(x_{5})F^{\mathcal{R}}_{\lambda\rho}\,U^{\dagger}\!(x_{5}) -\frac{1}{2}j^{+}_{\mu}F^{\mathcal{R}}_{\nu\sigma}\,U^{\dagger}\!(x_{5})F^{\mathcal{L}}_{\lambda\rho}\,U(x_{5}) \nonumber \\ &+i F^{\mathcal{L}}_{\mu\nu}\,j^{-}_{\sigma}j^{-}_{\lambda}j^{-}_{\rho}+i F^{\mathcal{R}}_{\mu\nu}\,j^{+}_{\sigma}j^{+}_{\lambda}j^{+}_{\rho} +\frac{2}{5}j^{-}_{\mu}j^{-}_{\nu}j^{-}_{\sigma}j^{-}_{\lambda}j^{-}_{\rho}\Bigl] \;\ \label{wzw} \end{align}$$