formal languages – Is string spliting formally defined when the string delimiter is an empty string?

Given sequences $A = (a_1, ldots, a_n)$ and $B = (b_1, ldots, b_m)$, write $A + B = (a_1, ldots, a_n, b_1, ldots, b_m)$ for their concatenation. Given a sequence of sequences $C = (C_1, ldots, C_n)$, write $Sigma(C) = C_1 + cdots + C_n$ for their concatenation.

Define a splitting of a sequence $X$ by a sequence $Y$ to be a sequence of sequences $Z = (Z_1, ldots, Z_k)$ such that $Z_i neq Y$ and $Z_i neq ()$ for all $i = 1, ldots, k$, and
$$X = Sigma (Z_1, Y, Z_2, Y, z_3, ldots, Y, Z_k).$$

For example, $((a), (), (b, c))$ is a splitting of $(a, u, v, u, v, b, c)$ by $(u, b)$.

Splitting by the empty sequence is not unique: $(a, b, c)$ may be split by $()$ in several ways, among others $((a),(b),(c))$, $((a,b), (c))$ and $((a,b,c))$. From a theoretical point this is a rather trivial and non-interesting observation. The implementors of various string libraries need to deal with splitting by the empty sequence somehow, and as you show, they do.