Serbia mobilized before answering the ultimatum and rejected the key demand. At that time Russia issued a pre-mobilization order.
That month before the start of the war was a time of great confusion in Austrian, German, and Russian governments, -- which all lacked adequate crisis management mechanisms. It is possible to find a quote from the Keiser, for example, supporting several contradictory positions, -- depends which minister he spoke to just before that.
I will refer to the Keegan's book and give a more substantive response later.
"Serbia mobilized before answering the ultimatum and rejected the key demand."
No.
Just, "No."
Serbia accepted Austria's first two demands and *suggested* that Austria's third demand be submitted for arbitration.
Austria didn't even reply, save to say that Serbia's response was inadequate, and Austria certainly didn't attempt to negotitiate. Austria simply invaded anyway, revealing their ultimatum to be a ploy in the first place.
Serbia's acceptance of Austria's demands was so well done that Austria's chief ally Germany, in the form of Kaiser William II, said that there was now "no excuse for war."
Austria invaded Serbia anyway.