Whichever entity directs that work should describe it: the duties, the duration, the supervision, and how it relates to the university role. A passing reference inside an academic appointment letter is usually too thin to explain an arrangement that involves a second organization. Ownership of a document follows the party whose conduct it proves. A letter about laboratory supervision belongs to whoever supervises; a letter about pay belongs to whoever pays. For each fact in dispute, ask which entity could be contradicted by its own records, and request the letter from that one.
Let each entity speak for its own work
If the institute is legally distinct, a university letter describing what happens inside the institute carries limited weight, because the signer is not the person who directs that work. Ask for a letter from the entity that does, covering the same ground as the academic one: role, duties, dates, and reporting line. Keep both, unaltered, in the same file.
Certain records settle points that letters only assert. A payroll register shows which entity actually compensates the professional; a signed services agreement shows which one directs the work; a degree transcript or an entry in a licensing body's register shows whether the qualification attached to the listed profession is met on its face. Where an outside administrator assembled the package, confirm each item came from the organization it purports to represent, because a compiled binder can carry an authority nobody granted it.