Street Demon 625 experience so far

That is the check valve. I had one that was stuck in a holley and the darn thing would not squirt. Its just a gravity check valve. The acc pump plunger may be a double spring loaded affair: The linkage acts on a rod that pushes a spring that acts on the plunger that has another return spring under it. This style prevents the linkage from "hydro-locking" and bending in the event of a plugged squirter and it also allows the pump plunger to provide a controlled push when the throttle is blipped and the pump cannot discharge fast enough to keep up with the mechanical linkage. Holley has them on the end of their acc pump arm too. Your "pushing on the pump rod" may have just been the first spring rebounding, against a stationary piston.