A high school band has purchased music with permission to perform. The music teacher has requested that the performance be shared on the school's website. From my understanding, the performance may be shared live / streamed (permission to broadcast) via the school's web page but may not be recorded and then posted to the website.
Your understanding is correct, but there are three additional details it is helpful to consider in this type of situation:
First, when a school confronts a concern like this, it should take a careful look at the license (the permission to use a copyright-protected composition) it purchased.