We say that the stationary time series $(X_t)_{t \in \mathbb Z}$, , with mean $\mu$, is mean ergodic if the following converge in probability holds: \begin{equation}\label{a}\tag{M-E} \hat{\mu}_T := \frac{1}{T}\sum_{t=1}^T X^t \overset{p}{\to} \mu \end{equation}
But, others define ergodicity when the converge in \ref{a} holds for other convergence. For example, here (slide 14) the author uses quadratic mean, and here (slide 5), almost sure convergence.
I know well that quadratic mean convergence and almost sure congergence, each one, implies convergence in probability. So I have no problem defining mean-ergodicity using quadratic mean convergence or almost sure.
The problem is when it comes to showing that a certain process is not mean-ergodicity. For example, in the same lecture (slides 19 and 20) given above, it shows that it is not mean-ergodic by showing that quadratic mean convergence does not hold. However, it may still hold convergence in probability. I didn't check if in this case the convergence in probability doesn't hold too, but wouldn't it be better to prove the non-convergence in probability to avoid these cases?
Which convergence to use when showing no-men-ergodicity?