Thanks Skydog / Hillrider. Actually, from photos of the IceToolz tool, I wonder if it's out of the same factory as the Spin Doctor one.

You're quite right, the Park flats are just 1". It's not unusual (ba-dada-dadada) but I was thinking that it since I don't already have the socket I'd be buying two tools where one would do. Now that I look, seems that many BB tools also use the same size, so a 1" socket could come in handy for more than one thing.