special relativity - Does a "midpoint between" 2 members of an inertial system belong to the same IS?

This answer to a recent question related to Einstein's thought-experimental definition of (how to determine) simultaneity contained the following statement:

Suppose two people, $C$ and $D$, stand equal distances from you and are known to pitch balls at exactly the same speed. With everyone standing at rest, $C$ and $D$ each toss you a ball. You get the ball from $C$ *before* the one from $D$. This is not a logical inconsistency. It simply means $C$ threw a ball *before* $D$ in your reference frame [emphasis added].

I believe that I understand the described setup and the conclusion ("It simply means $C$ threw a ball before $D$") as such.
But I (still) question whether it is necessary to add the qualification "in your reference frame" (above and beyond pointing out the relevant participants $C$ and $D$ themselves), or whether that is superfluous.

Since Einstein's definition essentially requires to identify a "midpoint between" the two participants under consideration who are at rest to each other (such as "between railroad ties $A$ and $B$", in the reference linked above; or "between pitchers $C$ and $D$" in the quoted statement) therefore I'd like to know:

Is it necessesary to refer to
"the midpoint between pitchers $C$ and $D$ in your (and $C$'s and $D$'s) inertial reference frame"
in order to identify it uniquely,

instead of simply referring to "the midpoint between ween pitchers $C$ and $D$" ?

Is there a "midpoint between pitchers $C$ and $D$ in an (inertial) reference frame other than your's (and $C$'s and $D$'s)" ?
And if so, is such a "midpoint between pitchers $C$ and $D$ in an (inertial) reference frame other than your's (and $C$'s and $D$'s)" different from "the midpoint between pitchers $C$ and $D$ in your (and $C$'s and $D$'s) inertial reference frame" ?

(If not, then the qualification "in your reference frame" is apparently superfluous; and, indeed, it would seem inappropriate and misleading to add such a qualification as if it were necessary.)