It's probably more accurate to say that inward-pointing vectors are those with positive $x^n$ component. In the given boundary chart, each element of $T_pM$ can be written $v = v^i\partial/\partial x^i$ (using the summation convention), and $v$ is inward-pointing if and only if $v^n>0$.
First of all, the concept of a "manifold" is certainly not exclusive to differential geometry. Manifolds are one of the basic objects of study in topology, and are also used extensively in complex analysis (in the form of Riemann surfaces and complex manifolds) and algebraic geometry (in the form of varieties).
Within topology, manifolds can be studied purely as topological spaces, but it is also common to consider manifolds with either a piecewise-linear or differentiable structure. The topological study of piecewise-linear manifolds is sometimes called piecewise-linear topology, and the topological study of differentiable manifolds is sometimes called differential topology.
I'm not sure I would necessarily describe these as distinct subfields of topology -- they are more like points of view towards geometric topology, and for the most part one can study the same geometric questions from each of the three main points of view. However, there are questions that only make sense from one of these points of view, e.g. the classification of exotic spheres, and there are certainly topology researchers who specialize in either piecewise-linear or differentiable methods. Differential topology can be found in position 57Rxx on the 2010 Math Subject Classification.
Differential geometry, on the other hand, is a major field of mathematics with many subfields. It is concerned primarily with additional structures that one can put on a smooth manifold, and the properties of such structures, as well as notions such as curvature, metric properties, and differential equations on manifolds. It corresponds to the heading 53-XX on the MSC 2010, and the MSC divides differential geometry into four large subfields:
Classical differential geometry, i.e. the study of the geometry of curves and surfaces in $\mathbb{R}^2$ and $\mathbb{R}^3$, and more generally submanifolds of $\mathbb{R}^n$.
Local differential geometry, which studies Riemannian manifolds (and manifolds with similar structures) from a local point of view.
Global differential geometry, which studies Riemannian manifolds (and manifolds with similar structures) from a global point of view.
Symplectic and contact geometry, which studies manifolds that have certain rich structures that are significantly different from a Riemannian structure.
As a general rule, anything that requires a Riemannian metric is part of differential geometry, while anything that can be done with just a differentiable structure is part of differential topology. For example, the classification of smooth manifolds up to diffeomorphism is part of differential topology, while anything that involves curvature would be part of differential geometry.
Best Answer
To really connect the claims I make below with the definitions given in your post takes some effort, but since you asked for intuition here it goes...
I think the best intuition for push-forwards and pull-backs is offered from the case $\mathcal{M}=\mathcal{N} $ and both are embedded in $\mathbb{R}^n$.
the push-forward amounts to changing coordinates for a vector field (viewed as a derivation) for example: for cartesians verses polars in $\mathbb{R}^2$ $$ \frac{\partial}{\partial r} = \frac{\partial x}{\partial r}\frac{\partial}{\partial x}+ \frac{\partial y}{\partial r}\frac{\partial}{\partial y} = \cos(\theta)\frac{\partial}{\partial x}+\sin(\theta)\frac{\partial}{\partial y}$$
the pull-back amounts to taking the total differential of the coordinate formulas, or equivalently, applying the chain-rule to swap differentials in one coordinate system for another. For example, again for two-dimensional Cartesian verse polars: $$ dr = d(\sqrt{x^2+y^2}) =\frac{x}{\sqrt{x^2+y^2} }dx+\frac{y}{\sqrt{x^2+y^2} }dy $$
The role of the pullback to integration is that it allows us to lift integration defined in $\mathbb{R}^n$ up to the manifold (provided we have the partition of unity to weave things together). Essentially the abstract $dx$ on the manifold pulls down to an ordinary differential on $\mathbb{R}^n$. However, it's not quite that simple because we have $dx \wedge dy = -dy \wedge dx$ on the manifold. In an integral on $\mathbb{R}$ we have $dxdy=dydx$ provided we swap the integration bounds appropriately. This apparent contradiction is reconciled by the fact that integrals of two-forms correspond to surface integrals. In a surface integral the order of the parameters determines the orientation of the surface. All things being otherwise the same $f(x,y)dx \wedge dy$ and $f(x,y)dy \wedge dx$ pull back to the same surface integral modulo a sign. Of course this sign is important, it measures outward or inward flux for the example I'm currently discussing.
All of this said, it bugs me to no end when texts choose to omit the $\wedge$ for explicit form calculations!
Per request: an explicit example Suppose we take $\phi (A) = (A_{11},A_{12},A_{21},A_{22})$ as the coordinate chart on $\mathcal{M} = \mathbb{R}^{ 2 \times 2}$. Denote the coordinate functions by $x_{ij}: \mathcal{M} \rightarrow \mathbb{R}^4$. To integrate in this 4-dimensional manifold we need to find some 4-form with which to play. Define, $$ \omega = (x_{11}^2+2)dx_{11} \wedge dx_{12} \wedge dx_{21} \wedge dx_{22} $$ To help separate concepts let me use $(u_1,u_2,u_3,u_4)$ as the standard cartesian coordinates on $\mathbb{R}^4$. Let's calculate the pull-back of $\omega$ under $\phi^{-1}=\Psi$. Since $\Psi^*(\omega)$ will be a four-form at $p=(p_1,p_2,p_3,p_4)$ on $\mathbb{R}^4$ it suffices to consider (suppressing the $p$ for now) $$ \Psi^*(\omega)(\partial_1,\partial_2,\partial_3,\partial_4)=\omega (\Psi_*(\partial_1), \Psi_*(\partial_2),\Psi_*(\partial_2),\Psi_*(\partial_4))$$ We need to calculate the push-forwards to continue, $$ \Psi_*(\partial_1) = \partial_{11}, \ \ \Psi_*(\partial_2) = \partial_{12}, \ \ \Psi_*(\partial_3) = \partial_{21}, \ \ \Psi_*(\partial_4) = \partial_{22} $$ Thus, \begin{align} \Psi^*(\omega)(\partial_1,\partial_2,\partial_3,\partial_4) &=\omega (\partial_{11},\partial_{12},\partial_{21},\partial_{22}) \\ &= (x_{11}^2+2)(dx_{11} \wedge dx_{12} \wedge dx_{21} \wedge dx_{22}) (\partial_{11},\partial_{12},\partial_{21},\partial_{22}) \\ &= x_{11}^2+2 \end{align} Following the $p$ through the calculation will reveal that we find: $$ \Psi^*(\omega)(\partial_1,\partial_2,\partial_3,\partial_4)_p = p_1^2+2 $$ Or, if you like, set $p=u$ and find $u_1^2+2$. To perform an integration we best take some compact subset of the matrix space then you can see that it will simply reduce to integrating the function $u_1^2+2$ on the corresponding parameter space in $\mathbb{R}^4$
About half way through this I realized you only wanted a covector example. For that, we simply need an abstract one-dimensional manifold and a one-form with which to play. If you wish I'll add that later.