In Hungary it's pretty simple: you're allowed to cross when the green man is visible. That's what that green man is supposed to signalize. That's its purpose. If it's red, you are not allowed to cross, because some other car traffic direction has priority.

Now of course you can cross in practice even if it's red for you, but it's illegal (rarely enforced).

So in the event of a red light for through traffic but a green light for turning traffic you don't want pedestrians crossing the road. That's why it is important to have separate lights indicating when a pedestrian may cross.

For instance, if I'm crossing the road here https://goo.gl/maps/favpGxyKTLk (this is on my daily cycle commute)

there is no way for me to know when there is a green arrow light for vehicles that come from opposite direction and turn left (and therefore come my way).

The buttons requesting green man for pedestrians and cyclists is, btw, known here as "humiliation button" (because it forces cyclists to stop; there is automatic detection for motor vehicles on the driveway, but for pedestrians and cyclists they haven't found a reliable way to do it).