I have a vague recollection that “nothing sucks like a vax” was a parody and never the vacuum cleaner’s slogan. A cursory google search doesn’t pull up anything authoritative and it seems like most of the claims that the vacuum cleaner people used it are all worded nearly identically. The photocopied afi saw back in the 80s was too on the nose to be really believable, feeling an awful lot like the mouse balls memo.
The ad I remember showed an actual vacuum cleaner. I'd seen the one you linked to in my cursory googling. I'd hoped it would have the same vacuum cleaner picture to make the identification that much easier.