Perhaps they are using puntalini for punterie.
Many older engine had pushrods and tappets.
If this is the case puntalini/punterie is tappets and aste will be pushrods imo.

This would also match the phrase Le aste non risultano essere deformate e i puntalini non usurati, as one of the things you'd be checking in an engine rebuild would be if the pushrods were bent and the tappets were worn.

tappet 
n.
(mecc.) punteria (del motore): tappet adjustment, registrazione delle punterie
● tappet rod, asta di rinvio □ tappet stem, asta della punteria.