The 4th paragraph from the Broadway League link you provided says:
Exceptions are being made for children under 12 and people with a medical condition or closely held religious belief that prevents vaccination. These guests must provide proof of a negative COVID-19 PCR test taken within 72 hours of the performance start time, or a negative COVID-19 antigen test taken within 6 hours of the performance start time.
And then on the MTC page you linked, the second paragraph under "MTC's Samuel J Friedman Theatre" says:
The only exception to the above will be for guests under the age of 12, or those who need reasonable accommodations due to a medical condition or sincerely held religious belief. Guests requesting an exemption must provide proof of at least one of the following, in addition to properly wearing a mask inside the building:
- Negative COVID-19 PCR test taken within 72 hours of the performance, or
- Negative COVID-19 rapid antigen test taken within 6 hours of the performance.
