My Dad had one, i THINK it was a Sears or possibly Montgomery Ward, less likely Western Auto. Except I kinda recall a brown or gold color - not sure.
I'm leaning towards a Wards but, it wasn't that much more convenient than a rope start, and the spring starter eventually broke. pretty sure it was a briggs engine though. I will NEVER forget that clicking sound of the spring starter! thanx for linking the video!
I also have vague memories of a gadget my dad got to chuck into the drill that had some 'nubs' designed to push into the perforated vent/screen on top of a mower to use the drill to start. Anyone recall that? maybe advertised in TV guide -- possibly on TV as well?