I've experienced plenty of beg buttons that don't do anything other than activate the crossing light when that direction next goes green for cars. But also plenty where pressing the button takes immediate steps for the pedestrian: if the direction is already green for cars, the walk sign is turned on; otherwise, the current cycle ends and any intermediate cycles proceed with minimum duration; non-dominant directions will have longer minimum cycle time if the pedestrian button has been pressed.

In my experience, if the intersection processes car traffic dynamically, it will also process pedestrian traffic dynamically, and generally favor pedestrians. If the intersection has static timing, it's not going to do anything different for pedestrians. I recall driving in Milwaukee near UWM two decades ago, every morning I could count on the lights showing exactly the same thing at the same time. If I left my apartment at the same time, I'd get the same sequence of missed or made lights. Most other places I've driven had car sensors (usually loops in the pavement, but cameras are cheaper and less maintenance now), so there's no schedule to be learned, although the behavior is predictable if you learn the cycles and observe the stimuli. (Although, sometimes sensors are broken or miswired; there was an intersection near one of my employers where the cyclist button was connected as an left turn lane occupancy sensor; resulting in consistently wrong behavior. it's also not uncommon for a disabled vehicle to be located above a loop, or sometimes a construction / utility vehicle to be parked above a loop; some control systems can adapt and filter that out, but many don't.)