Astrom-Hagglund Relay PID tuning - relay transfer function?
Your second PlantInput description is correct.

Relay tuning relies indeed on input steps whose sign depend on the sign of the error. An IEEE Control System Society file also graphically shows it. 

The diagram you posted is ambiguous, but interpreting the relay block as a sign function rather than a gain (which is usually of triangular shape) would dissolve your confusion and have it agree with your intuition.