I've never heard of a definition that requires an algorithm to be recursive.

I'd suspect that the original text has got something to do with recursive function theory. In that sense, an algorithm is recursive, since solving a problem involves:
moving from the intial condition to an intermediate state
moving from the intermediate state to another intermediate state (between 0 and a finite number of times
moving to the final state

Obviously, this is a recursive structure, which only bottoms out when you get to the atomic operations of the language.

--
Tommy
Too stupid to live.
Too stubborn to die.