I think what is confusing you is that the US Navy issued an Technical Specification order for design and development of a new fighter to certain specifications. Grumman, Curtiss, Bell, Republic, Vought, etc… were competing (or expected to design and offer bids). Grumman went ahead and designed, built and tested an aircraft XF6F-1 with the R2600-16 Wright engine and Curtiss electric prop. first flight: 062642, delivered to Navy:040243. This was the basis for the production F6F-3’s with the addition of the R-2800-10 and Hamilton Standard prop. The XF6F-2 had the Wright R-2600-15 & -21 plus Birmann turbo-supercharger in an accessory compartment, later delivered to US Navy as F6F-5 production model (engine changed to Pratt R-2800-10 built in 2 stage supercharger). XF6F-3 was the production prototype with R-2800-10 and 13’ Hamilton Standard constant speed prop. first flight: 073042; delivered 031543.
Production F6F-3 first flight: 100342, first delivery: 100442, last delivery: 040944. 4402 built for US Navy.
Production F6F-5 first delivery: 042944, last delivery: 111645. 7870 built for US Navy.
From a conversation with Dave Thurston (Chief Designer-project engineer on F6F series) Grumman built the first 2 prototypes without contracts, however as Grumman was a proven and preferred US Navy contractor the expenses were covered by the production contracts. I have no doubt that the US Navy did give the nod to officially sanction prototype builds in 061941. After all they would likely require several aircraft to begin operational flight testing and carrier trials. Remember these aircraft fully loaded were about twice the weight of fully loaded F4F’s. Arresting gear both aircraft borne and carrier deck, plus support equipment needed to be analyzed or developed, etc… Also there were the huge tasks of writing, publishing, verifying all the pertinent operational manuals, service training, parts inventories to establish, etc… The official contracting departments in the US Government and US Navy were worlds unto themselves, not unlike the legal beagles rooting around in today’s corporate hall ways… the real work is often long done before they saunter in to tell you " Okay go ahead". (sorry, I could not resist a quick parry with the rapier!)