The wording not “real” damage on page 146 of the PH is only used to distinguish Nonlethal Damage from Lethal Damage.
Lethal Damage, of course, is the most common category of damage in the game. Like Lethal Damage, Nonlethal Damage refers to a creature’s Hit Points but works differently. In order to make sure everyone understands how to handle Nonlethal Damage – it is not deducted from the current HP, instead it is counted separately and it also can’t kill you - the authors of the PH probably thought it’s a good idea to talk of not “real” damage. But we should acknowledge that they put the word real in quotation marks.
Nevertheless, Nonlethal Damage is damage.
damage: A decrease in hit points, an ability score, or other aspects
of a character caused by an injury, illness, or magical effect. The
three main categories of damage are lethal damage, nonlethal damage,
and ability damage. (...)
PH, 307 (emphasis mine)
The condition under which a protean scourge’s Split ability triggers is: "When damaged…“ Form or type of damage is not specified. Thus Nonlethal Damage will do to make a protean scourge split.
(which means, by RAW, a protean scourge would also split if it took Nonlethal Damage from starvation or drowning or any other circumstances that deal Nonlethal Damage. So maybe one should rule that the damage that triggers the split ability has to come from an attack...)