Or the annoying behaviour that there's no 'automatic mode' in peak hour, where there's a whole bunch of pedestrians waiting to cross on both sides every phase and everyone else assumes someone else has pushed the button, then the lights change and there's no green man...

Why do you need a green man to cross when you have a green traffic light?

Different places have different practices and different legislation. In some places, walking at a red light is forbidden and there could even be a fine. In some others, your red man indicates that even if there is a green light for some vehicles (going straight ahead), there might also be a green arrow light that tells turning vehicles have right of way. The green man tells you that you can go.

It's easy enough to see if there's a green arrow. Is that the actual problem? Or is every other signal identical to the situation where you're supposed to cross, except for the one on the crosswalk itself?

The lights are turned towards the cars. Sometimes it's physically impossible for you to see them, because they are on the other sides of roads, not visible from the side where you are. Also, even if you see the light, it might change rapidly and then you don't have enough time to make it across.

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).

Because of shades surrounding the traffic lights (to prevent glare from the sun) it is often impossible for pedestrians to see if a turn signal is lit from their perpendicular position.

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.

Right - and even without any shades, there are places where I can't see at all the light that permits vehicles to cross my path.

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).

When there's no junction, you don't, and in all cases green for traffic implies green for people - but not vice versa. At a crossroads traffic must be blocked in both directions for it to be safe to cross. A red light for traffic in one direction means vehicles travelling in the other direction could still turn left or right through it.

Hmm... green for one implies red for the other, is what I meant to say...

Maybe I should have added a disclaimer: I am not your driving instructor

"cross pedestrian inhibit"

As the are in affect two different crossing, with a stagger between them. The guidance is to avoid a pedestrian having to wait a whole cycle in the middle or tempted to rush both parts. One cycle wait, breaks down a bit for heavy used crossing. But the cycle time should be lower, if ped flow are up.

In an ideal situation, you could set the order of each part to be tidal - if near for example a station exit.