It matters as evidence. Payment and supervision arrangements are part of how a placement is judged to be a controlled service rather than labour for hire, so the commercial terms carry legal weight.
The contract is evidence, not just commerce
Hypothetical example: Filing fees are published in the USCIS fee schedule and are a small part of the picture. The financially significant point is that the structure chosen for commercial reasons will later be read as evidence. Where the client pays the petitioner for a defined deliverable, the petitioner pays and supervises the specialist, and the petitioner carries responsibility for the outcome, the arrangement looks like a service.
Where the client effectively directs the person, sets their hours and absorbs them into its own teams, the arrangement looks like labour for hire regardless of how the invoice is worded. A Selwyn firm negotiating a commissioning contract should involve the immigration question at the drafting stage, because the clause that would have solved it is much cheaper to insert before signature than to explain afterwards.