>>283546688
It's neither. It's one of those stupid things that requires Japanese context to understand.
See, the cuckoo bird they're talking about is sometimes called the 賤の田長 in Japan, which means VERY loosely "bird tells commonfolk it's rice planting season" (or more literally, the "peasant's rice field boss"), since it was known to mainly sing during that time of year. It's pronounced "shizu no taosa".
Eventually, probably though people saying its name wrong, the 賤/shizu (peasant) part shifted into 死出/shide (die/travel to the afterlife). So people started calling the bird 死出の田長 (rice field boss of death). With a name like that, people naturally started treating it as if it were a messenger from the afterlife.
Meanwhile, Buddism treated the journey to the afterlife (死出) as this divine thing that involved a journey of fire and some shit. So when the story of the Phoenix found its way from Egypt to China, they ate that shit up and the two things were connected pretty quick. So there were a fuckton of Buddist poets who treated the Phoenix as a metaphorical guide to the afterlife.
So in Monogatari, the phoenix who guides people to the afterlife and the cuckoo that was known as a messenger for the afterlife somehow got conflated into an oddity called 死出の鳥 (death/afterlife journey bird). It's not strictly a phoenix or cuckoo, just a weird thing that came into existence because there were two birds associated with 死出. So it's kind of both and neither at the same time.
Also fuck NISIO.