Actually, they changed the belt (and its associated device) midway into the 518 run. Prior to mid-1958 the belt to be used was part #1590 (same soft belt as used for the upright brush on models 505-515), but with the new "Mark II" design a dedicated belt, part #405058, was to be used.
BELT USAGE
Mark I Handi-Butler (models 510 to early 518): part no. 1590 (NLA, use part no. 159056)
Mark II Handi-Butler (later model 518 to 561): part no. 405058 (NLA)
Mark III Handi-Butler (562 Sanitronic to Dual Sanitronic 80): part no. 405058 (NLA)
Mark IV Handi-Butler (Classic, Classic Omega): part no. 405058 (NLA)
Mark V Handi-Butler (Classic III, Tradition, Heritage): part no. 405058 (NLA)
~Ben