The admin for the style is not in the ACP as the style states.
The Bridge of Hope style is designed for any type of charity or band style websites. It includes the Vertex Framework version 2.0 that includes the frontend admin capability
I did click in the Style Admin. That second screenshot is not in my forum at all. Which means no bright blue buttons.
I see what's going on. When I click on Style Admin like you suggest, I'm taken to the moderator control panel. There is no Vertex control panel in here.