What might be the definition of a positively oriented chart in From Calculus to Cohomology?

A very long question!

As you know, the concept of orientation arises in linear algebra by taking equivalence classes of ordered bases of a real vector space $V$, two such bases $\{b_i \}$ and $\{b'_i \}$ being equivalent if the linear automorphism sending $b_i$ to $b'_i$ has positive determinant. There are exactly two orientations of a vector space $V$ with dimension $> 0$. For a general $V$ none of these two orientations is privileged and it would be an arbitrary choice to call one of them positive and the other negative. However, if $\omega$ is an orientations of $V$, it makes sense to write $-\omega$ for the other orientation, i.e. the minus-sign indicates that the orientation is reversed. Note that a linear isomorphism $f : V \to W$ between vectors spaces $V,W$ establishes a bijection between ordered bases of $V,W$, and thus between orientations of $V,W$. We can therefore say that linear isomorphisms transfer orientations between vectors spaces.

In contrast to the general case, $\mathbb{R}^n$ as the standard model of an $n$-dimensional real vector space has a canonical ordered basis $\{ e_1,\dots,e_n \}$, and its equivalence class is customarily denoted as the positive orientation of $\mathbb{R}^n$. This special situation is due to the fact that the set $\{ 1,\dots,n \}$ has a natural order.

There are various equivalent approaches to define the concept of an orientation on a differentiable manifold $M$. In my opinion the best approach is to define an orientation of $M$ to be a family $\Omega = (\omega_p)_{p \in M}$ of compatible orientations of the tangent spaces $T_pM$. But what is meaning of compatible? The problem is that $T_{p_i}M$ are distinct for $p_1 \ne p_2$, thus we cannot say that the orientations $\omega_{p_i}$ of $T_{p_i}M$ agree.

Let us first consider the simple case of an open subset $V \subset \mathbb{R}^n$. The tangent spaces $T_xV$, $x \in V$, are all distinct, but there is a canonical linear isomorphism $h_x : T_xV \to \mathbb{R}^n$. This allows to define an orientation of $V$ to be a family of orientations $(\omega_x)_{x \in V}$ of orientations of $T_xV$ such each each $x_0 \in V$ has an open neigborhood $V_{x_0} \subset V$ such that for each $x \in V_{x_0}$, $h_x$ transfers $\omega_x$ to the same orientation of $\mathbb{R}^n$. It is easy to see that a connected $V$ has exactly two orientations. We can moreover say that an orientation of $V$ is positive if each $h_x$ transfers $\omega_x$ to the positive orientation of $\mathbb{R}^n$. Finally, if $R : \mathbb{R}^n \to \mathbb{R}^n$ is a reflection at a hyperplane, e.g. $R(x_1,\dots,x_n) = (-x_1,x_2\dots, x_n)$, then we see that the diffeomorphism $R_V = R : V \to R(V)$ has the property $h_{R(x)} \circ T_xR_V = -h_x$, i.e. $R_V$ is orientation reversing.

An orientation of a differentiable manifold $M$ is now defined as a family of orientations $\Omega = (\omega_p)_{p \in M}$ of $T_pM$ such that for each chart $\phi : U \to V \subset \mathbb{R}^n$ the family $\phi_*(\Omega) = (T_{\phi^{-1}(x)}\phi(\omega_{\phi^{-1}(x)})_{x \in V})$ is an orientation of $V$. The chart $\phi$ is said to be positively (negatively) oriented with respect to $\Omega$ if $\phi_*(\Omega)$ is the positive (negative) orientation of $V$. Obviously each chart on a connected $U$ is either positively or negatively oriented. If $U$ is not connected, we can only say that the restriction $\phi_\alpha$ of $\phi$ to each component $U_\alpha$ of $U$ is either positively or negatively oriented. Moreover, for each chart $\phi : U \to V$ there exists a chart $\phi' : U \to V'$ such that $h_{\phi'(p)}(\phi'_*(\omega_p)) = - h_{\phi(p)}(\phi_*(\omega_p))$ for all $p \in U$ (simply take a reflection $R : \mathbb{R}^n \to \mathbb{R}^n$ and define $\phi' = R_V \circ \phi : U \to V' = R(V)$). Working componentwise, we see that on each chart domain (which is an open subset $U \subset M$ which occurs as the domain of a chart) there exist both positively and negatively oriented charts.

The collection of all positively oriented charts forms an atlas for $M$. All transition functions between charts in this atlas have the property that the sign of the determinant of the Jacobian matrix is $+1$ at each point. Note that the collection of all negatively oriented charts has the same property.

Any atlas having the above property it called an orientable atlas, and this is an alternative way to introduce the concept of orientation on manifolds.

Note, however, that there are no open subsets $U \subset M$ which are positively oriented in an absolute sense: Positive orientation is a property of charts with respect to an orientation $\Omega$.