Let's try it with x = 2, y = 5:

    S(2) add 5       (i.e. 3 + 5)
     => 2 add S(5)   (i.e. 2 + 6)
     => S(1) add 6   (i.e. 2 + 6)
     => 1 add S(6)   (i.e. 1 + 7)
     => S(0) add 7   (i.e. 1 + 7)
     => 0 add S(7)   (i.e. 0 + 8)
     => 0 add 8
     => 8            (base case)

I agree that your rule would work too though:

    S(2) add 5
     => S(1 add 5)
     => S(S(0 add 5))

Seems more sensible to me, and by adding the same rule to the right hand side we end up only needing a single base case (that 0+0=0)

     => S(S(S(0 add 4)))
    ...
     => S(S(S(S(S(S(S(S(0 add 0))))))))
     => S(S(S(S(S(S(S(S(0))))))))
         = 8 (that is the definition of 8)