"to busy oneself in a busy or ineffective way" is the second definition of "putter" in Webster's Newworld Dictionary 1985. I'm too lazy to look in my 1962. This was published in the US, and have always lived in the central US.
I'm equally comfortable with both 'putter' and 'potter', among other words for the same sort of activity.
My experience as well. I've always thought "potter" came first, and "putter" came from an alternate pronunciation. And bother are fine.
And in fact (, Anne!), M-W.com says taht this "putter" was na alteration of "potter" that first appeared in print in 1858.
Potter comes from "pote" --to poke--and has been around since 1829.