The leader, Silver, literally said that "those who White does not trust, we do not trust." That denotes some ability that White has to be able to tell who is good or bad (at least to me).
The medal from the tower signifies her as king Salomon's companion.
Soooooo. You missed the entire point of where she only revealed the medal AFTER they summarized their entire organization's internal structure. (〜￣△￣)〜
Well he did tell his companion that if you want something you have to give something when he asked why you told so much.
... What kind of organization essentially tells a stranger you only met once their whole organization's internal structure? (￣∇￣") I mean sure the stranger saved one of your members, but for all they could know, MC might have been collaborating with their enemies....