Definition of Manifold's Orientation

Question

I am reading the book of manifold. And I find there are many definitions about one object, such as orientation, Euler character and degree of map.

I am confused with the conception of orientation. This may be related to https://mathoverflow.net/questions/10966/two-kinds-of-orientability-orientation-for-a-differentiable-manifold but different.

The first definition is from Lee's Introduction to Topology Manifold.

First definition Let $K$ be $n$-dimensional simplical complex. An orientation of $K$ is a choice of orientation of each $n$-simplex in such a way that two simplices that intersect $(n-1)$-face are consistently. If a complex $K$ admits an orientation, it is said to be orientable.

The second definition is from Greenberg's Algebraic Topology: A First Course.

Second definition Let $M$ be $n$-dimensional manifold. $S:M\rightarrow \cup_{p\in M}H_n(M,M-p)$ which satisfies:

(1)$S(p)$ is a generator of $H_n(M,M-p)$.

(2)For any $p\in M$, there exist $p\in U$ and $\sigma$ is a generator of $H_n(M,M-x)$: for any $q\in U$, we have $S(q)=j_q(\sigma)$.

So my question is: are the two definitions same? Can anyone give some details? Thank you.