A topological $n$-manifold is a second countable Hausdorff space such that every point has a neighbourhood which is homeomorphic to an open ball centred at the origin in $\Bbb{R}^n$.
The "centred at the origin" part is equivalent to other formulations of the definition of a manifold, as can be clearly inferred.
When talking about homeomorphisms between neighbourhoods of points and open balls in $\Bbb{R}^n$, I always visualize the point around which the neighbourhood is constructed to map to the origin. Is this intuition correct?
If it is, then I have the following doubt:
In a manifold with boundary, every point has a neighbourhood which is homeomorphic to the $n$-dimensional upper half space, or $\Bbb{H}^n$. The boundary of $\Bbb{H}^n$ is the set of points where $x_n=0$. If $M$ is a manifold with boundary, a point that is in the inverse image of $\partial \Bbb{H}^n$ under some chart is called a boundary point of $M$.
Isn't every point in $M$ a boundary point of $M$ then? How does that make sense?