Is pink eye considered a "pink eye" when no itching or pain?

Yes. Pink eye is just an eye that is pink for any reason. Pink eye could be an infection, in which case there is often pus. (this kind is contagious.) it may be allergy in which case it is usually itchy. (not contagious) it may be trauma in which case there is often pain. It may be irritation in which case there may be pus, itching and/or pain.
Yes. "pink eye" is the description of the whites of the eye, usually in the vernacular of a viral infection. However, pink/ red eyes can be caused by many different things: infections, inflammations, dryness, allergies, chemicals, trauma to name some. If your eyes are asymptomatic but pink, you could be tired or dry or exposed to the elements of the local environment.