Confusion about whether a fermion field and its conjugate as an Grassmann number

I'm confused about what "a Grassmann-odd number" really means and how does it apply to fermions.

In some text, it says that "if $\varepsilon \eta+\eta \varepsilon =0 $, then $\varepsilon $ and $\eta$ are Grassmann-odd numbers.

And in wiki, the Grassmann algebra are those whose generators satisfy $$\theta_1\theta_2+\theta_2\theta_1=0$$and fermion space is one of the Grassmann algebras.

So, following these definitions, a fermion field is a Grassmann-odd number because$$\{\psi_\alpha,\psi_\beta\}=0$$so is its conjugate. But$$\{\psi_\alpha,\psi^\dagger_\beta\}=\hbar\delta_{\alpha \beta}.$$So why don't $\psi_\alpha$ and $\psi^\dagger_\beta$ anticommute if they are all Grassmann-odd number?

I suppose that if they are not belonging to the same Grassmann algebra. But if so, when the essay says "$\epsilon$ is a Grassmann-odd number" (e.g.Chapter II of an essay about supersymmetry), what does it mean?