faithfulness of SO(3)-SU(2) and euler angles

Question

I know from theory that the adjoint representation $\tau_a$ of group $\mathcal G$ is faithful iff $\mathcal G$ is centerless. From basic theory it turns out that parametrizing a rotation in $\mathbb R^3$ with direction $\hat{n}$ and angle $\alpha$ ($\boldsymbol{\alpha}=\alpha\hat{n}$) the adjoint representation is not faithful because if $\boldsymbol{\alpha}'=-\hat{n}\frac{2\pi-\alpha}{\alpha}$ we have $\tau_a(\boldsymbol{\alpha}')=\tau_a(\boldsymbol{\alpha})$. On the other hand we can set a correspondence with the elements of the fundamental representation of SU(2) (in which we choose a parametrization consistent with SO(3)'s one) $\tau_f$ and we find that $\tau_f$ is faithful. On the lecture notes i'm reading of QFT (http://www.robertosoldati.com/archivio/news/107/Campi1.pdf) on page 31 it says that with euler angles the situation is reversed with $\tau_a$ faithful and $\tau_f$ not. If a representation (for a Lie group) is a map from the group manifold $\mathcal G$ to $Aut(V)$, with V linear space, how should a property of the representation depend on the parametrization of the manifold? I'm a bit confused, am I missing something that this appears to me so strange or i have just misunderstood something?