I'm in such a highly developed Metropolitan area that it doesn't even get dark at night. You see my tagline, "In the Shadow of The Big Chicken!", well Interstate 75 is in between my home and The Big Chicken. I'm three and a half miles from Dobbins Air Reserve Base. I walk up my driveway at night and see a glow to the southwest as bright as the dawn.
The only thing around here that I'm aware of other than White Water Park is the Olympic Park downtown which is just fountains that people can walk through. As I said initially, I have never heard of a splash pad.
So I googled it and I guess it's a new age term that my Boomer hears have not been exposed to because there are water parks in the Metro area that have "splash pads", so I guess I can still learn something after all.
Wait a minute, the one here is named "City of Marietta Elizabeth Porter Park & Sprayground", so I'm vindicated slightly. As you first questioned, "(i)s it that they’re named differently in different places or simply don’t exist?" The one in my city is named differently, so the former rather than the latter.
I think I have heard “spray park”, a variation of that.