Orientation-compatible intersections

Question

(Note: Cross posted on MO in https://mathoverflow.net/questions/426436/orientation-compatible-intersections since not clear what is a suitable platform)

This question is about clarifications on the following definitions from "Necessary and sufficient conditions for hyperplane transversals" by Pollack and Wenger and "Hadwinger's transversal theorem in higher dimensions" by Goodman and Pollack.

1. The $d$-order type of an ordered set of points $P = \{ p_1, \ldots, p_n \} \subset \mathbb{R}^d$ is defined to be the family of signs of $\det \begin{pmatrix} 1 & p_{i_0}^1 & \cdots & p_{i_0}^d \\ \vdots & \vdots & \cdots & \vdots \\ 1 & p_{i_d}^1 & \cdots & p_{i_d}^d \end{pmatrix}$ for $1 \le i_0 < \cdots < i_d \le n$. If some subset $Q \subset P$ of the points spans a linear subspace of dimension $k < d$, then the $k$-order type of $Q$ is said to be determined by the $d$-order type of $P$.

2. A $k$-ordering of a family of sets $\mathcal{B} = \{ B_1, \ldots, B_n \}$ is defined via a corresponding set of points $P = \{ p_1, \ldots, p_n \}$ in $\mathbb{R}^k$. The orientation of a $(k + 1)$-tuple of $\mathcal{B} = \{ B_{i_0}, \ldots, B_{i_k} \}$is defined as the orientation of the corresponding $(k + 1)$-tuple of points $\{ p_{i_0}, \ldots, p_{i_k} \}$.

3. An oriented $k$-linear subspace $F$ intersects a subset $\mathcal{A}$ of $\mathcal{B}$ consistently with a given ordering of $\mathcal{B}$ if we can choose from each $B \in \mathcal{A}$ a point in $B \cap F$ schedule that the chosen points have the same order type as the corresponding points in $\mathbb{R}^k$.

Questions:

1. What is the exact definition of an oriented $k$-plane? Does this depend on a specific choice of ordered generators?

2. In part 3, does consistency of intersection depend on something other than the ordering of the designated intersection points (e.g. how we obtain points in $\mathbb{R}^k$ or the matrix formed by the coordinates of these vectors)?

3. There is also a version of part 3 (intersections consistent with an orientation) for oriented matroids ("Oriented matroids and hyperplane transversals" by Anderson and Wenger). What would be a precise definition of this? Are the points in question now the ordered set of vectors used in the chirotope or covector definitions of an oriented matroid?