The spectrum of a ring minus a prime of height $1$

Question

Let $R$ be a ring (commutative, with unit) and let $Q$ be the localization of $R$ at its regular elements (non zero divisors). Let $\mathfrak{p}$ be a prime ideal in $R$.

Let $R[\mathfrak{p}^{-1}]\subset Q$ be the subring of $Q$ containing $R$ defined as: $$ R[\mathfrak{p}^{-1}]:= \{r\in Q~|~\exists n\geq 0,~\forall p\in \mathfrak{p}^n,~px\in R\}. $$ It is the union of the $\mathfrak{p}^{-n}R$ in $Q$. The canonical map $R\to R[\mathfrak{p}^{-1}]$ induces a morphism of spectra: $$ \iota:\operatorname{Spec}R[\mathfrak{p}^{-1}] \longrightarrow \operatorname{Spec}R. $$ I was wondering in which cases we can claim that $\iota$ induces an isomorphism $\operatorname{Spec}R[\mathfrak{p}^{-1}]\cong (\operatorname{Spec}R)\setminus V(\mathfrak{p})$ of ringed spaces. This does not happen very often:

1. If $\mathfrak{p}=(0)$, then $R[\mathfrak{p}^{-1}]=R$ and thus $\iota$ is the identity but $(\operatorname{Spec}R)\setminus V(\mathfrak{p})=\emptyset$.
2. As in @reuns's example (in comment), consider $R=\mathbb{C}[x,y]$ and $\mathfrak{p}=(x,y)\subset R$. $V(\mathfrak{p})$ is closed in $\operatorname{Spec} R=\mathbf{A}^2_{\mathbb{C}}$, and $(\operatorname{Spec} R)\setminus V(\mathfrak{p})$ is a scheme, obtained by glueing of $\operatorname{Spec} \mathbb{C}[x,y,x^{-1}]$ and $\operatorname{Spec} \mathbb{C}[x,y,y^{-1}]$ along $\operatorname{Spec} R$. It follows that $$ (\operatorname{Spec} R)\setminus V(\mathfrak{p})=\operatorname{Spec}\mathbb{C}[x,y,x^{-1},y^{-1}]. $$ On the other-hand, we have an inclusion $\mathbb{C}[x,y,x^{-1},y^{-1}]\subset R[\mathfrak{p}^{-1}]$ which is strict, as $(x+y)^{-1}$ belongs to latter and not to the former. Therefore, $\iota$ induces: $$ \operatorname{Spec}R[\mathfrak{p}^{-1}] \longrightarrow (\operatorname{Spec}R)\setminus V(\mathfrak{p}) $$ but it is not an isomorphism of ringed spaces.
3. If $R$ is a Dedekind domain and $\mathfrak{p}$ is not zero, then we have $R\cap \mathfrak{p}R[\mathfrak{p}^{-1}]=R$, and if $\mathfrak{q}\neq \mathfrak{p}$ is a prime of $R$, then $R\cap \mathfrak{q}R[\mathfrak{p}^{-1}]=\mathfrak{q}R$. In particular, $$ \operatorname{Spec}R[\mathfrak{p}^{-1}] \longrightarrow (\operatorname{Spec}R)\setminus V(\mathfrak{p}) $$ is an isomorphism of ringed spaces.

EDITED: My question is the following. If $\mathfrak{p}$ has codimension/height $1$, does $\iota$ induces an isomorphism $\operatorname{Spec}R[\mathfrak{p}^{-1}]\cong (\operatorname{Spec}R)\setminus V(\mathfrak{p})$ of ringed spaces?

Many thanks!