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.)

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.

This has already been addressed and answered despite your evident refusal to accept that it has.

There are circumstances in which you, at the midpoint and in the same IRF as the pitchers, receive the ball from C first but, as observed from another IRF, C does not throw the ball first.

Thus, the qualification "in your reference frame" is not necessarily superfluous.

Observe (from Wiki): "Events A, B, and C occur in different order depending on the motion of the observer. The white line represents a plane of simultaneity being moved from the past to the future."

• Alfred Centauri: "[...] but, as observed from another IRF, [...]" -- Please try to directly address the stated question(s) and/or to directly refer to the definition in the linked reference. Are you really suggesting that there exists (at least) one member of another IRF (other than the IRF of $C$ and $D$) who should be called "midpoint between" (or "middle between") $C$ and $D$ ??, in order to give judgements about simultaneity (or sequence) specificly of $C$'s and $D$'s acts in the sense of Einstein's definition. If so: who? (in terms of $\beta$, distance $CD$ etc.) – user12262 Oct 17 '13 at 22:10
• @user12262, are you deliberately being obtuse? Your argument is flawed. Isn't it clear? It does not logically follow from your midpoint "argument" that the qualification in your reference frame is superfluous. I've already demonstrated that the qualification is not superfluous thus, your argument is flawed - period. – Alfred Centauri Oct 17 '13 at 22:21
• @user12262, to be more specific, isn't it clear, that the quoted, highlighted phrase in your question, that in your reference frame qualifies the phrase "C threw a ball before D" and not anything at all to do with the midpoint? You seem to be obsessively focused on the the midpoint for no reason at all. – Alfred Centauri Oct 17 '13 at 22:39
• Alfred Centauri: "that in your reference frame qualifies the phrase "C threw a ball before D"" -- Agreed; obviously. "and not anything at all to do with the midpoint" -- ??? Of course it does: The term "before" in the quoted phrase requires a definition that (to quote the reference) in every real case it must supply us with an empirical decision as to whether or not the conception that has to be defined is fulfilled. Therefore: an observer [should be] placed at the mid-point M of the distance [$CD$]. (Clearly, defining "sequence" is closely related to defining "simultaneity".) – user12262 Oct 17 '13 at 23:02
• @user12262, the answer to the question of whether C throws first or not, as observed in an IFR, does not require that there be an observer at the midpoint - the midpoint observer is utterly irrelevant and obviously a major impediment to your understanding of the physics involved here. The answer is determined by the reading of Einstein synchronized clocks co-located with each pitcher when each throws the ball. No middleman required. – Alfred Centauri Oct 18 '13 at 0:04