Of course I would accept their account of events in your example, but this is a far way from the demand by protesters for a blanket "believe the victims". This demand IS a call for guilty until proven innocent.There's a world of difference between "believing" someone and coming to a legal verdict.
It's quite possible and even reasonable to believe someone without any evidence whatsoever. Sometimes our personal judgements are based purely on trust, not evidence.
For instance if your wife or partner came home in a highly distressed state telling you they'd been sexually assaulted, would your response be, "I'm sorry love but I'm afraid I can't believe you until I've heard your alleged assaulter's version of events."