Here are some concrete examples which may shed some additional light on the issue. Let's consider the manifold $\mathcal M = \mathbb R \times \mathrm S^1$, i.e. the cylinder. Points $p\in \mathcal M$ can be labeled by a triple $(z,a,b)$, where $z\in \mathbb R$ and $a^2+b^2=1$. It's critical to note that $(z,a,b)$ are not the coordinates of $p$ in some chart, since $\mathcal M$ is a 2-dimensional manifold.
A local, active transformation is a diffeomorphism $\phi : U \rightarrow U$, where $U\subset \mathcal M$ is some neighborhood. In our case, let's choose
$$U := \big\{ (z,a,b) \in \mathbb R \times \mathrm S^1 \ \bigg| \ a > 0\}$$
$$\phi : U \ni (z,a,b) \mapsto (z+1,a,b)$$
In words, $\phi$ just pushes the points of the manifold along the cylindrical axis by one unit.
A local, passive transformation is a change of chart. Let $x$ be a chart map defined on the subset $U$ as before, defined by
$$x:(z,a,b) \mapsto \big(z, \tan^{-1}(b/a)\big)\in \mathbb R^2$$
Now let $y$ be a different chart map, defined by
$$y : (z,a,b) \mapsto (z^3, \tan^{-1}(b/a)\big)\in \mathbb R^2$$
The chart transition map $(y \circ x^{-1}):(z,\theta) \mapsto (z^3 ,\theta)$ maps from the $x$ coordinates to the $y$ coordinates. However, this is not a map from $U\rightarrow U$; the points $p\in U$ aren't actually going anywhere, we're just choosing different labels for them.
A global, active transformation is a diffeomorphism $\Phi: \mathcal M\rightarrow \mathcal M$. As an example, we could let
$$\Phi :\mathcal M \ni (z,a,b) \mapsto (z,-a,-b)$$
A global, passive transformation is a change of chart where each chart covers the entire manifold. One such chart is the following:
$$ x : \mathcal M \rightarrow \mathbb R^2-\{(0,0)\}$$
$$ (z,a,b) \mapsto (e^za, e^z b)$$
Another example would be
$$ y: \mathcal M \rightarrow \mathbb R^2-\{(0,0)\}$$
$$(z,a,b) \mapsto (-e^z b, e^z a)$$
The chart transition map is $(y \circ x): (\alpha,\beta) \mapsto (-\beta,\alpha)$, which corresponds to a $90^\circ$ rotation. Once again, note that this is not actually moving points in $\mathcal M$ around; it's just changing the labels.
Finally, a Lorentz transformation is one which preserves the Minkowski metric. An active Lorentz transformation is a (global or local) diffeomorphism which is also an isometry of the Minkowski metric; a passive Lorentz transformation is a (global or local) change of chart which preserves the form of the Minkowski metric, i.e.
$$\frac{\partial x^i}{\partial y^a} \frac{\partial x^j}{\partial y^b} \eta_{ij} = \eta_{ab}$$