Form 337
There seems to be some confusion on when/how the 337 is used. In the US, most 337s are used as part of a 'Field Approval'. In this way, the FAA FSDO inspector approves the data that is used for the modification (this could be a follow-on from an existing STC that isn't directly applicable to the subject aircraft), or it could be the manufacturer's installation manual. The FSDO inspector will make a statement in box 3 confirming approval of the data. One the installation has been carried out, the repair station or IA will make the box 7 approval for return to service after the A&P or repair station has cleared box 6 for conformity of the modification to the apprpoved data, which will be listed on the back page of the 337.
In the alternative case, the installation data will already be approved by an STC (this may be an AML STC with hundreds of aircraft listed such as for the JPI EDM700), or it could be data approved by a DER on an 8110-3. In this situation, the IA or repair station will just sign the box 7 and there will be no FAA entry in box 3. Once the aircraft is released to service via the 337, the completed form is sent to FAA in Oklahoma, not to the FSDO.
Therefore, in all cases, the 337 must be completed and signed with approved data of some form BEFORE the aircraft can fly. You can't submit an un-approved form to a FSDO or ACO and believe that you can fly whilst waiting for it to be signed off.
For a minor change, the 337 is not used. A minor change just requires a logbook entry.