The proof of Proposition 15.9 from John Lee's book "Introduction to Smooth Manifolds" is left as an exercise.
Here is the statement:
Let $M$ be a connected, orientable, smooth manifold with or without boundary. Then $M$ has exactly two orientations. If two orientations of $M$ agree at one point, they are equal.
Here is my argument
Let $\mathcal{O}=\{\mathcal{O}_p:p\in M\}$ and $\tilde{\mathcal{O}}=\{\tilde{\mathcal{O}}_p:p\in M\}$ be orientations for $M$.
Let $\mathcal {C}=\{p\in M:\mathcal O_p=\tilde{\mathcal O}_p\}$, and let $p\in \mathcal C$.
By definition there exist $(E_i:U\to TM)$ and $(\tilde E_i:\tilde U\to TM)$ (continuous) local frames such that $p\in U\cap \tilde U$ and $(E_i)$ is positively oriented with respect to $\mathcal O$ and $(\tilde E_i)$ is positively oriented with respect to $\tilde {\mathcal O}$.
We can suppose $U\subseteq \tilde U$ and $U$ connected. Since $\mathcal O_p=\tilde{\mathcal O}_p$ we have that the ordered basis $(E_1|_p,\dots,E_n|_p)$ and $(\tilde E_1|_p,\dots,\tilde E_n|_p)$ are consistently oriented, meaning that the transition matrix $A(p)=(A_i^j(p))$ has positive determinant.
The map det$_A:U\to \mathbb{R}, q \mapsto$det$A(q)$ is continuous (where $A(q)$ is the transition matrix between the ordered basis $(E_1|_q,\dots,E_n|_q)$ and $(\tilde E_1|_q,\dots,\tilde E_n|_q)$) and since $U$ is connected and det$A(p)>0$ we have that det$_A$ is always positive on $U$. This implies that $U\subseteq \mathcal C$, and thus $\mathcal C$ is open in $M$.
Analogously we show that $M-\mathcal C$ is open in $M$.
Since $M$ is connected we have $\mathcal C=\emptyset$ or $\mathcal C=M$. In the second case we have $\mathcal O=\tilde{\mathcal O}$. In the first case $\mathcal O$ and $\tilde{\mathcal O}$ are two distinct orientations of $M$ and any other orientation $\hat{\mathcal O}$ of $M$ would have $\hat{\mathcal O_p}=\mathcal O_p$ or $\hat{\mathcal O_p}=\tilde{\mathcal O_p}$, and thus we would have $\hat{\mathcal O}=\mathcal O$ or $\hat{\mathcal O}=\tilde{\mathcal O}$. $\qquad\square$
To be precise I should also prove the existence of two distinct orientations. But if $\mathcal O$ is the orientation which exists by hypothesis, then $-\mathcal O$ is another orientation. So we have at least two (distinct) orientations of $M$.
Please let me know if my proof is correct and if it can be shortened/ simplified.