Stage Manager/Production Manager/Director/Producer are a slightly different kettle of fish (being management)
Not true - SM and Director are also independent contractors. PM might be a staff/management position, but might also be a contractor (happens rarely). Producer certainly is a managing figure.
Please note too that if you make other recordings for broadcast - not for archival purposes (for example: radio (NPR), television (PBS), or satellite (simulcasting)), you must have an additional agreement with your rights provider and each individual union involved in your production. Usually these union agreements will stipulate that you must pay the artists an additional fee for any broadcast/recorded performance. Then you will have to get each artist to sign a contract addendum. It's a lot of legwork, but that's how you cover your bases.
Listen, most publishers or unions aren't going to file a lawsuit against your neighborhood community theatre for somebody's mom filming her kid's performance as "2nd monkey from the right". Nor do they send a secret agent representative to be sure that no spy cameras are recording the performance. Do your best to try not to let it happen, but don't run around screaming "lawsuit". The big thing is that having illicit recordings out there will weaken your position when you go to negotiate your new union contract.