Shell we call the first just arahat or ariya perspective and the second one - more mundane, in your words - puthujjana perspective?
Is it not our job to change our perspective from the second one to the ariya perspective? One pretty famous theravada monk and scholar during "dyeing"process was shouting "I am dying, I am dying". Surely at that time he saw "self" or more simply himself as a subject to death.
Yes, I think Buddha can say :"I am old" and this is probably your mundane sense. The problem is so puthujjana really believes "I am so and so", I was born and I will die. Otherwise Buddha would not put so strong emphasis on disidentification "This is not mine, this I am not, this is not my self". Who sees dependent arising sees Dhamma, so dependent arising is the most important perspective to acquire and in this perspective death is impermanent, conditioned and dependently arisen.
Once I believe I have mastered a truth, the argument which interests me is not the one which confirms it but the one which refutes it.
A confused idea attracts a fool like a flame attracts an insect.
Nicolás Gómez Dávila