Time lines of observers meeting each other - doubts about their graphical representation I am trying to make sense of Fig. 2.12, page 23 of Introducing Einstein's Relativity (D'Inverno, Oxford University Press). There it goes:

The book picture is in black. Scales are such that light rays are inclined by 45deg.
Observer A sees events P and Q happen simultaneously at equal opposite distance. 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.
This does not convince me. In my view an observer that meets A the very moment A observes P and Q must be travelling on the RED line.
The book says that A sees P, Q and O happen at the same time. I'd say that A sees P, Q and O' happen at the same time.
Which is the correct time line for B? The black or the red one?
 A: 
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.
A: The complete diagram from d'Inverno (p. 23) is shown below.
It appears that observer-A has performed radar experiments on events P and Q. Observer-A assigns time-coordinates to events as the halfway time between emission and reception of the radar signal. So, Observer-A assigns P and Q the same time coordinate---to Observer-A, P and Q are simultaneous. Further, it appears that event O is the midpoint-event between emission and reception, and thus O is simultaneous with P and Q.
Granted, at the meeting event O, observer-A doesn't yet have the information needed to assign those time-coordinates to P and Q yet. 
As others have pointed out, "sees" (or observes) is an imperfect term since one might not distinguish (1)
a spacelike-relation of simultaneity with [i.e., assigning the same t-coordinates to] a distant event
from (2)
a past-lightlike-relation with [i.e., visually seeing] a distant event.
Referring to your statements:


*

*"A sees P, Q and O happen at the same time." really means that A assigns the same time-coordinate to P, Q, and O.

*"A sees P, Q and O' happen at the same time." would be correct if it means that light-signals from P, Q, and O' reach A at the same time... according to A. If you said, with the meaning just given, "A sees P, Q and O' at the same event", then everyone would agree to that.
B's worldine passes through event O, as the book has drawn.
The purpose of the diagram is that Observer-B will assign distinct time-coordinates to events P, O, and Q.

