You are 100% correct, there is no dump truck license, most places need you to be 'competent' which is doubled up for a while an familiar with the site etc, then assessed before you head out on your own
How do I know? I'm in HSE, and have worked on different sites and worked with a lot of different people
Drivers license is only for public roads
Having said that, a lot of companies are expecting employees to have tickets to operate - same sort of deal as loaders etc - you only need a license to drive them on public roads, but then they need to be registered , which some of the larger plant cannot be licensed or registered for road use
Really, a drivers license is just a 'ticket of competency' and arguably that considering some drivers.
Bottom line - if you have a ticket, it may help