Two kinds of orientability/orientation for a differentiable manifold

If $X$ is a differentiable manifold, so that both notions are defined, then they coincide.

The ``patching'' of local orientations that you describe can be expressed more formally as follows: there is a locally constant sheaf $\omega_R$ of $R$-modules on $X$ whose stalk at a point is $H^n(X,X\setminus\{x\}; R).$ Of course, $\omega_R = R\otimes_{\mathbb Z} \omega_{\mathbb Z}$.

This sheaf is called the orientation sheaf, and appears in the formulation of Poincare duality for not-necessarily orientable manifolds. It is not the case that any section of this sheaf gives an orientation. (For example, we always have the zero section.) I think the usual definition would be something like a section which generates each stalk.

I will now work just with $\mathbb Z$ coefficients, and write $\omega = \omega_{\mathbb Z}$.

Since the stalks of $\omega$ are free of rank one over $\mathbb Z$, to patch them together you end up giving a 1-cocyle with values in $GL_1({\mathbb Z}) = \{\pm 1\}.$ Thus underlying $\omega$ there is a more elemental sheaf, a locally constant sheaf that is a principal bundle for $\{\pm 1\}$. Equivalently, such a thing is just a degree two (not necessarily connected) covering space of $X$, and it is precisely the orientation double cover of $X$.

Now giving a section of $\omega$ that generates each stalk, i.e. giving an orientation of $X$, is precisely the same as giving a section of the orientation double cover (and so $X$ is orientable, i.e. admits an orientation, precisely when the orientation double cover is disconnected).

Instead of cutting down from a locally constant rank 1 sheaf over $\mathbb Z$ to just a double cover, we could also build up to get some bigger sheaves.

For example, there is the sheaf $\mathcal{C}_X^{\infty}$ of smooth functions on $X$. We can form the tensor product $\mathcal{C}_X^{\infty} \otimes_{\mathbb Z} \omega,$ to get a locally free sheaf of rank one over ${\mathcal C}^{\infty}$, or equivalently, the sheaf of sections of a line bundle on $X$. This is precisely the line bundle of top-dimensional forms on $X$.

If we give a section of $\omega$ giving rise to an orientation of $X$, call it $\sigma$, then we certainly get a nowhere-zero section of $\mathcal{C}_X^{\infty} \otimes_{\mathbb Z} \omega$, namely $1\otimes\sigma$.

On the other hand, if we have a nowhere zero section of $\mathcal{C}_X^{\infty} \otimes_{\mathbb Z} \omega$, then locally (say on the the members of some cover $\{U_i\}$ of $X$ by open balls) it has the form $f_i\otimes\sigma_i,$ where $f_i$ is a nowhere zero real-valued function on $U_i$ and $\sigma_i$ is a generator of $\omega_{| U_i}.$

Since $f_i$ is nowhere zero, it is either always positive or always negative; write $\epsilon_i$ to denote its sign. It is then easy to see that sections $\epsilon_i\sigma_i$ of $\omega$ glue together to give a section $\sigma$ of $X$ that provides an orientation.

One also sees that two different nowhere-zero volume forms will give rise to the same orientation if and only if their ratio is an everywhere positive function.

This reconciles the two notions.

Your main question was answered by Emerton. Regarding other notions of orientability, there's many. A popular one is the obstruction-theoretic approach:

1) A manifold $M$ is orientable if the tangent bundle $TM$ admits a trivialization when restricted to a $1$-skeleton of a CW-decomposition of $M$. An orientation of $M$ is taken to be a (homotopy class of) trivialization of $TM_{|M^0}$ that extends over $M^1$.

2) [Corrected to take into account Chris's comment] You can restate definition 1 in a way that avoids skeleta. A popular one is to define the associated orthogonal (principal) bundle to $TM$, lets call it $O(TM)$. This is the bundle over $M$ whose fibers over points $p \in M$ is the linear isomorphisms between $\mathbb R^m$ and $T_pM$. Then $M$ is orientable if every loop $S^1 \to M$ lifts to a loop $S^1 \to O(TM)$.

3) There's a small variant on these ideas called the "orientation cover", this is a 2-sheeted covering space of $M$, and it is connected if and only if $M$ is non-orientable. This has the additional assumption that $M$ is connected.

4) Another variant on this comes from bundle classifying-space machinery. Every vector bundle has a classifying map $M \to B(GL_m)$, and $GL_m$ has a subgroup of positive-determinant matrices, call it $GL^+_m$. $M$ is orientable if and only if the classifying map $M \to BGL_m$ lifts to a map $M \to BGL^+_m$, and an orientation is a homotopy-class of such lifts (flexible enough to allow homotopy of the original classifying map).

Anyhow, those are a few. There's of course more since all these ideas admit perturbations in various directions. For example, another small variant would be that the 1st Stiefel-Whitney class is trivial. One advantage to approaches (1), (2), (4) is that any of them are natural lead-in to other notions of orientation, like $spin$ or $spin^c$ structures.

Also I ask, are there any additional ways to define orientability/orientation for a differentiable manifold(not just for a vector space)?

Another notion of orientability is the existence of an atlas whose transition functions have derivatives with everywhere positive determinant. This gives a clear cut way, along with the Cauchy-Riemann equations, of showing that every complex manifold (say, for simplicity, a Riemann surface) is orientable.