My question is whether the below notion of powers of ideals, which may be regarded as a weakening of the notion of symbolic power, has been already defined in the literature and, in that case, I ask for a reference.

Let $R$ be a commutative Noetherian regular local ring, and let $I\subseteq R$ be a radical ideal. Fix a non-negative integer $n$.

If $I=\cap p_j$ is the primary decomposition of $I$, then set
$$
I^{<n>}:=\{f\in R:\ f^n\in p_j R_{p_j} \text{ for all } j\}.
$$
Geometrically, I think this corresponds to all the sections that vanish generically on $V(I)$ at order $n$, where $V(I)$ denotes the algebraic set defined by $I$.

Notice that $I^{<n>}$ contains the symbolic power $I^{(n)}$ and, in particular, contains the ordinary power $I^n$; on the other hand, the inclusion can be strict, just take $I=(xy,xz,yz)$; the element $xyz$ belongs to $I^{<2>}$, but not to $I^2$, neither to $I^{(2)}.$ On the other hand, if $I$ is generated by a regular sequence, then $I^{<n>}=I^{(n)}=I^n$.