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.
Best Answer
I just wanted to mention that while orientability for cohomology with arbitrary coefficients is governed solely by cohomology with coefficients in ℤ, there are other cohomology theories for which is is not true. For example, if you have an action of $\pi_1(X)$ on an abelian group M, then you can talk about (co)homology with twisted coefficients in M. For any vector bundle there is a coefficient module such that the bundle is orientable with respect to these twisted coefficients (or, to paraphrase Matthew Ando, "every bundle is orientable if you're twisted enough").
Also, one can ask whether a vector bundle is orientable with respect to topological K-theory, real or complex, or many other generalized cohomology theories, which capture interesting information about the manifold.
So while ℤ/m-coefficients may not be the most interesting coefficient systems to study orientability in, they're part of a larger systematic family of questions (and they don't take much extra work if you're already doing ℤ and ℤ/2-coefficients).
Finally, for something like real coefficients you might think of orientations that differ by a real scalar geometrically, e.g. according to some volume form.