Local Definiteness and Primary state

Question

In "Local Quantum Physics" by R. Haag[H1], the Local Definiteness is defined as folllwing:

Let $\mathcal{O}_n$ be a directed set of regions shirinking to a point $x$, i.e. $\mathcal{O}_{n+1}\subset \mathcal{O}_n;\quad\cap \mathcal{O}_n=x$. Any two physically realizable states $\omega_i(i=1,2)$ will become indistinguishable in restriction to $\mathfrak{A}(\mathcal{O}_n)$ in the limit $n\rightarrow \infty$. Specifically \begin{equation} \|(\omega_1-\omega_2)|_{\mathfrak{A}(\mathcal{O}_n)}\|\rightarrow 0\text{ as }n\rightarrow \infty. \end{equation}

Here $\mathfrak{A}$ is a net of algebra. In various other materials, i.e. for example [V], it defines the Local definiteness as following.

Local Definiteness: $\cap_{\mathcal{O}\ni p} \mathcal{R}_\omega(\mathcal{O})=\mathbb{C}\mathbb{1}$ for all points $p$ in the sapcetime and $\omega\in \mathcal{S}$.

We can consider $\mathcal{S}$ as the set of physically realizable states. Here $\mathcal{R}_\omega(\mathcal{O})\equiv \pi_\omega(\mathfrak{A}(\mathcal{O}))''$, where $\pi_\omega$ is a GNS-representation of state $\omega$.

I have three questions.

1. In another paper [H2], it says:

   (...) it means that the algebra of a compact, contractible region shall be a $W^*$-factor, i.e. isomorphic to an operator algebra with trivial center on a Hilbert space and that only normal states of this algebra are allowed. We shall call this the principle of local definiteness.

   But I think this rather refers to the Local Normality condition in [H1], which is:

   In restriction to the algebra of a finite, caontractible region all physically realizable states belong to a unique primary folium.

   And (if I understood correctly) this condition comes when we assume "(...)that among the realizable states we have states which are primary for the algebras of small regions(...)", which definitely is a stronger condition. Buy they are written by the same authors. Am I missing something, or are these terms are used in a quite vague way?
2. Related to the question above, in [H1] Haag says the following.

   If we believe, in addition, that among the realizable states we have states which are primary for the algebras of small regions then, appealing to theorem 2.2.16 we conclude from the principle 3.1.3 that for any two folia of physically allowed states and any point $x\in \mathcal{M}$ there will be a neighborhood of $x$ such that the restrictions of the folia to the algebra of this neighborhood coincide and are primary.

   Theorem 2.2.16 says that two disjoint states has norm distance $2$; Principle 3.1.3 is the Local Definiteness. I understood that if Principle 3.1.3 is accepted, then any two physically realizable states have a neighborhood $\mathcal{O}$ of $x$ such that they are not disjoint in the algebra $\mathfrak{A}(\mathcal{O}).$ Then I am stuck. If we focus on the primary states, then two primary states are disjoint xor quasi-equivalent[Corollary 8.22 of [K]), thus we get the desired conclusion. But I cannot approach further.
3. Is the definitions in [H1] and [V] coincides? I actually caonnot even say that one implies the other. Maybe the Fell's Theorem[F] helps, but I don't know how can I apply it.

References:

[F] Fell, James MG. "The dual spaces of *-algebras." Transactions of the American Mathematical Society 94.3 (1960): 365-403.

[H1] Haag, Rudolf. Local quantum physics: Fields, particles, algebras. Springer Science & Business Media, 2012.

[H2] Haag, Rudolf, Heide Narnhofer, and Ulrich Stein. "On quantum field theory in gravitational background." Communications in Mathematical Physics 94 (1984): 219-238.

[K] Landsman, Klaas. Foundations of quantum theory: From classical concepts to operator algebras. Springer Nature, 2017.

[V] Verch, Rainer. "Local definiteness, primarity and quasiequivalence of quasifree Hadamard quantum states in curved spacetime." Communications in Mathematical Physics 160.3 (1994): 507-536.