How do I set what "maximize"means for a window?
the maximise feature on OS X is set to in some apps like Terminal (because it's content is purely text and can easily by formatted) to fit your whole screen, on other apps like Safari or Chrome, because they are browsers they tend to contain images (and other hidden HTML or CSS contents) which would make it by default to arrange the window so that it displays all the content correctly but also to allow you to work at the same time with other windows, that's why it will try not to use all the width (people tend to use 2 windows sideways not upways) but remember that (at least Safari) tends to remember the height and width that you manually chose.

On the 3 step cycle I think there is no way to do that, sorry, what I usually do is usually set browsers to use all the space available, if I need to use another app, either enter fullscreen mode with Safari and you can easily change from desktop to fullscreen apps with (control) + (left or right arrow). If you really want to do something like comparisons and really need to have both windows opened at the same time us El Capitan's split screen feature, set Safari (or others) to fullscreen and then access expose and drag your other windows to Safari.

Hope this helps,

Tiferrei