David, I came across the chronicle in another context. The original text is the Itinerarium peregrinorum et gesta Regis Ricardi, probably written or compiled by a monk at the end of the 12th century (very close to the events described).
The Latin text can be found in the Gallica database
http://gallica.bnf.fr/ark:/12148/bpt6k50246v.r=Itinerarium+regis+ricardiIt seems to me that the translator into English in the 19th century interpreted the Latin word "ballista" with "sling" every time, no matter if it made sense or not. Hence the passage with Richard the Lionheart using a "sling" with arrows which later turns out to be an arbalest. Obviously, the word ballista in the Latin original text is mostly used for the crossbow (arcuballista), maybe also for small siege engines, but probably not for the sling.
It would be interesting, though, to scan through the original to find if the "real" sling, funda, is also mentioned. There are many combat and siege scenes described.