The OP has a very important point.
While the vagueness of the language in the so called "contract" benefits GSE and I understand why it's there It should be changed. Definitions are in order here.
And she is right that it should never had taken a supervisor to get involved on something this obvious.
Training of these call center people seems to be a persistent issue in most of these complaints.
My road service is coming up for renewal soon (Aug) and while I have thankfully not had the opportunity to use the service I do keep an eye on this forum and I do worry that if needed I might find myself in a similar situation.