Fig. 2.12, page 23 of Introducing Einstein's Relativity (D'Inverno, Oxford University Press).
Observer A sees events P and Q happen simultaneously
This formulation constututes imprecise use of the notion of "simultaneity" according to Einstein's definition in two seprate senses. Instead:
(1): Observer A sees events P and Q "at the same time", i.e. in coincidence, at (only) one event (called O' in the sketch).
(2): The notion of simultaneity is not even applicable to entire events (but instead applicable to indications of individual observers, who took part in events; but who, unfortunately, are not explicitly drawn in the figure as reproduced above.
at equal opposite distance.
Distances are defined and to be determined between suitable pairs of observers (namely specificly: between observers who were and remained at rest between each other); not between an observer (such as A) and an event (such as P, or Q).
Again: Unfortunately, the figure doesn't show or name a specific observer who was and remained at rest wrt. A and who took part in event P (although such an observer might be easily considered in the setting of a thought-experment, let's call it J); nor another specific observer who was and remained at rest wrt. A and who took part in event Q (although such an observer might be easily imagined, too, let's call it K).
According to the book, observer B (riding his own BLACK time line) meets A at the same moment as events P and Q happen according to A.
Again: that's a misattribution of simultaneity to entire events. What can be said correctly and according to everyone involved, in mutual agreement, is instead:
Observer A's indication of having met observer B (namely at event O) was simultaneous to observer J's indication of having taken part in event P, as well as simultaneous to observer K's indication of having taken part in event Q.
In my view an observer that meets A the very moment A observes P and Q must be travelling on the RED line. [...] I'd say that A sees P, Q and O' happen at the same time.
Correct: Observer A observed all that in coincidence.
It is important to note the distinction:
The event in which A indicated perceiving events P and Q and being met (in passing) by the red-line-observer in coincidence (namely event Q)
is not the same as
the event (O) in which observer A took part such that A's indication at this event (specificly: A's indication of having met, in passing, by observer B) was simutaneous to observer J's indication of having taken part in event P, as well as observer K's indication of having taken part in event Q.