Oh gosh! I think I read it too quickly, so more to the topic:

  "...In a situation where you agreed to cover a portion of the expenses, would you use the structure."

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   I think that these descriptions fit, I hope I have not misunderstood! anyway; if not the case, please you all feel free to correct or comment them.

   Payment per service,   pay per services provided, 

   FEE-FOR-SERVICE (payment)
A traditional system of payment to providers in which providers are paid their costs along with a profit margin for rendering services. Under this system a fee is charged for each service provided on a retrospective basis. This means that after the services are rendered the fee is determined as compared to a prospective, fixed rate basis.
   (I lost the source...)