I want to address an ambiguity around the use of the word 'disagree' here.
I propose the following about thought demonstrations:
all of the protagonists featured in the narrative are to be physicists who are fluent in application of relativistic physics.
In a thought demonstration of relativity of simultaneity the two observers are both aware of relativity of simultaneity. Both observers will agree that the point of view of the other observer is self-consistent.
So: in that sense the two observers do not disagree with each other.
The actual message is: in Minkowski spacetime simultaneity is underdetermined.
As pointed out in a comment by stackexchange contributor Dale, the limits are that the same causality relations must still obtain.
Within the boundaries of still obtaining the same causality relations there is a certain leeway in adopting a plane of simultaneity.
Within that leeway Einstein synchronization procedure is one of the available options. For a given inertial coordinate system the Einstein synchronization procedure is the symmetrical approach. As a general rule, when there is leeway, pick the symmetrical choice as your convention.
(You do need to pick one choice, in order to represent physics taking place. Once you have made your pick - for a given inertial coordinate system - you must stick with that choice, otherwise you would introduce self-contradiction.)
To the two observers:
The two observers have a velocity relative to each other, so for each the co-moving inertial coordinate system is a different one. For each inertial coordinate system the Einstein synchronization procedure arrives at a different plane of simultaneity.
As stated earlier, we should assume the two protagonists of the thought demonstration are professional physicists, so they are aware of the reason why for two different inertial coordinate systems the Einstein synchronization procedure arrives at two different planes of simultaneity.
So that is why I find it awkward to see assertions that the two observers will disagree. Disagree about what? We should assume both are professional physicists: they will not disagree about the proper application of special relativity.
As pointed out in other answers: 'relativity of simultaneity' refers exclusively to that what remains after transmission delays have been taken into account.
(Therefore, if you read a discussion of relativity of simultaneity, and the author suggests that transmission delay effects are part of the story of relativity of simultaneity, then stop reading that author.)
The series of animated gif's by Andrew Hamilton for special relativity, including discussion of relativity of simultaneity