What exactly is meant by saying that two events had been "simultaneous in an inertial frame"?

Question

In order to address my question based on a concrete example setup let the following two separate events be given:

• participants $A$ and $J$ encountering each other in passing (additional participants may be considered). The details of this event $\varepsilon_{AJ}$ were widely observable: participant $A$'s indication "$A_J$" as a distinctive "sparkle", and participant $J$'s indication "$J_A$" as a distinctive "flicker",

• participants $B$ and $K$ encountering each other in passing (additional participants may be considered). The details of this event $\varepsilon_{BK}$ were widely observable: participant $B$'s indication "$B_K$" as a distinctive "twinkle", and participant $K$'s indication "$K_B$" as a distinctive "flash".

Let participants $A$ and $B$ having been and remained at rest to each other (but not to participants $J$ or $K$); and (likewise) participants $J$ and $K$ having been and remained at rest to each other (but not to participants $A$ or $B$).

Let participant $M$ be identified as "(located at the) midpoint between" $A$ and $B$ (for short: as "the middle between" $A$ and $B$), and let participant $N$ be identified as "the middle between" $J$ and $K$.

Further, it is given that participant $M$ observed (the signal fronts of) (all relevant details of) both events, $\varepsilon_{AJ}$ and $\varepsilon_{BK}$, in coincidence;
and on the other hand that participant $N$ first observed (the signal fronts of) (all relevant details of) event $\varepsilon_{AJ}$, and only subsequently event $\varepsilon_{BK}$.

It can be concluded that

(1) participants $A$, $B$ and $M$ were jointly members of an inertial frame (in the sense of Rindler),

(2) participants $J$, $K$ and $N$ were jointly members of an inertial frame,

(3) participant $M$ was not identifiable as "the middle between" $J$ and $K$,

(4) participant $N$ was not identifiable as "the middle between" $A$ and $B$,

and following Einstein's coordinate-free definition of how to determine "simultaneity" that

(5) participant $A$'s indication $A_J$ and participant $B$'s indication $B_K$ were simultaneous, and

(6) participant $J$'s indication $J_A$ and participant $K$'s indication $K_B$ were not simultaneous.

Now, a phrase which seems related and which is encountered quite frequently is that

"(two) events were simultaneous in a (particular inertial) frame"

and I'd like to understand whether and how this phrase might be used in case of the described example setup. Therefore:

Two specific Questions:

Regarding the given example setup would it be considered appropriate to say:

"events $\varepsilon_{AJ}$ and $\varepsilon_{BK}$ were simultaneous in the inertial frame of participants $A$, $B$, $M$"

?

And if so:
Is thereby anything implied besides the conclusions (1) -- (6) listed above, or even contradicting these conclusions (1) -- (6)
(for instance, might be implied that it is appropriate to say:

• "participant $M$ was the middle between $J$ and $K$, in the inertial frame of participants $A$, $B$, $M$", or

• "participant $J$'s indication $J_A$ and participant $K$'s indication $K_B$ were simultaneous in the inertial frame of participants $A$, $B$, $M$", or

• "participant $N$ was the middle between $A$ and $B$, in the inertial frame of participants $J$, $K$, $N$", or

• "participant $A$'s indication $A_J$ and participant $B$'s indication $B_K$ were not simultaneous in the inertial frame of participants $J$, $K$, $N$"

)
?

Post scriptum
for ready reference, and in response to comments by @Angelika :

The wording of Einstein's coordinate-free definition of "simultaneity" reads:

[An] observer placed at the mid-point $M$ of the distance $AB$ [...] If the observer perceives the two flashes of lightning at the same time, then they are simultaneous.

Albert Einstein, Relativity: "The Special and General Theory", Section 8: "On the Idea of Time in Physics" (1917), translated by R. W. Lawson

In application of this definition to the example setup described in the OP above therefore

• participant $M$, being the "middle between" participants $A$ and $B$, has the privilege and business of judging whether and which indications of $A$ and of $B$ were "simultaneous", or not; and likewise

• participant $N$, being the "middle between" participants $J$ and $K$, has the privilege and business of judging whether and which indications of $J$ and of $K$ were "simultaneous", or not.

$M$'s verdict concerning specificly $A$'s "sparkle" indication $A_J$ and $B$'s "twinkle" indication $B_K$ is:
these two indications were simultaneous; as listed in (5) above.
$N$'s verdict concerning specificly $J$'s "flicker" indication $J_A$ and $K$'s "flash" indication $K_B$ is:
these two indications were not simultaneous; as listed in (6) above.

Answer 1

I would like to bring the ladder paradox here to explain simultaneity of events.

A ladder (an inertial frame) is moving horizontally with a relatively high constant speed with respect to a garage (another inertial frame). The garage has an open door where the ladder can not actually enter if the ladder was at rest in the garage's frame but that is not important. Now the two events:
1. First end of the ladder touches first side of the door.
2. Second end of the ladder touches second side of the door.
In the garage's frame of reference, the ladder undergoes a Lorentz length contraction such that the above two events happens simultaneously while in the ladder frame of reference, the door width undergoes a Lorentz length contraction and the above two events doesn't happen simultaneously.
So, whether two events are simultaneous depends on your choice of reference frame.
I can now think in this way:
$A$ and $B$ are standing at the first and second sides of the door respectively of the garage while $M$ is at the middle of $A$ and $B$ in the garage reference frame and $J$ and $K$ are standing at the first and second end of the ladder respectively while $N$ is at the middle of $J$ and $K$ in the ladder reference frame. So, the answers to your specific questions are:
events $ε_{AJ}$ and $ε_{BK}$ were simultaneous in the inertial frame of participants $A, B, M$ since all the indications reached $M$ at the same time in the inertial frame of participants $A, B, M$ but the events were not simultaneous in the inertial frame of participants $J, K, N$ since the indications reach $N$ at different times in the inertial frame of participants $J, K, N$.

Answer 2

Regarding your assertions:

Events $\varepsilon_{AJ}$ and $\varepsilon_{BK}$ were simultaneous in the inertial frame of participants $A$, $B$, $M$.

This is a perfectly reasonable statement and it is the sort of language used in everyday physics.

Participant $M$ was the middle between $J$ and $K$, in the inertial frame of participants $A$, $B$, $M$.

This is not entirely a reasonable statement, though, because in the inertial frame of $A$, $B$ and $M$, $M$ can only be in the middle of $J$ and $K$ at a single specific time, so you'd need to specify that. A complicated and technical way to phrase it would be

• There exists an event $\varepsilon_M$ at which $M$ observes herself to be equidistant and collinear with $J$ and $K$; this event is simultaneous with $\varepsilon_{AJ}$ and $\varepsilon_{BK}$ in the inertial frame of participants $A$, $B$, $M$.

Less technically, you could say that $M$ passes through the middle of $J$ and $K$ at the same time as the flashes and sparkles happen, as observed in the frame of $A$ and $B$.

Note that terms like "collinear" and "equidistant" are tricky to define, because you're specifying distances to other objects which may be moving with respect to you, so you need to specify equidistance with respect to what times of observation - i.e. equidistance to events, not objects, where you observe the events as simultaneous.

Your third statement, on the other hand,

Participant $J$'s indication $J_A$ and participant $K$'s indication $K_B$ were simultaneous in the inertial frame of participants $A$, $B$, and $M$

is perfectly fine.

Your last two statements,

Participant $N$ was the middle between $A$ and $B$, in the inertial frame of participants $J$, $K$, $N$

Participant $A$'s indication $A_J$ and participant $B$'s indication $B_K$ were not simultaneous in the inertial frame of participants $J$, $K$, $N$

behave the same as the previous two.