How does this recurrence relation fit the algorithm?
You should be careful about what is being counted. If you want to count "runtime" (as in, number of statements) the given recurrence is clearly wrong.

Reverse-engineering from the recurrence, it is probably supposed to count additions. In particular, note how the computation the parameters for recursive calls is ignored. I hope it is clear that the recurrence is correct for the worst-case then; your reasoning for $+1$ is spot-on.

Why does restricting your self to additions make sense here? Additions are one of the statements that are executed most often (in the worst case, that is ignoring line 4.). That means that the true runtime (in a theoretic model) is proportional to the number of additions. Since you are (most likely) after asymptotics in terms of $O$, counting additions is sufficient: the proportion-constant is lost in $O$, anyway. 

You can, for the fun of it, set up a recurrence counting all statements. You'll see that the result has the same asymptotic growth as the one you were given.