There are many formulas for charging for design services. You’ll want to find someone who is completely transparent and open about this topic.
I charge a set hourly rate. The hours depend on the requirements and complexity of the project. The fee only increases if the project increases. In my practice, every client receives a detailed design proposal outlining the scope of the project prior to me beginning work. In most cases a contract is signed and I require a retainer with the balance due once I’ve completed the design tasks outlined in the proposal.
I always put a line item in my design proposals about billing for additional time with either the client or on the client’s behalf. This is for work not included in the proposal or for meetings not specified in the proposal.
It is both the designer’s responsibility to present a clear proposal and the client’s responsibility to make sure they understand the proposal and that it covers all the expectations they have for the designer’s scope of work.