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. The public performance relates to the site/building and not to the World Wide Web.
Please confirm whether my understanding is correct.
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.
This is because a license for sheet music can convey not only permission for on-site performance and broadcast, but also "recording" and "publishing" (posting). I have observed that the range of these permissions will vary not only between publishers, but even between songs at the same publisher. So, before recording (or deliberately not recording), check the fine print; you might have more (or less) permission than your district thought.
Second, it is good to consider why the school wants to make the recording and post it on the school website. Is it to simply showcase the band on a page dedicated to the school's achievements? Is it for fundraising purposes? Or is it posted as part of a student newspaper or student club newscast? If the post is part of an academic endeavor--especially one related to commentary or gathering news--posting part of a recorded performance could be a fair use.
And third--though still on the topic of fair use--it is important to remember that "Circular 21" pertaining to "Reproduction of Copyrighted Works by Educators and Librarians" confirms that the Copyright Act allows educators to make:
"A single copy of recordings of performances by students...for evaluation or rehearsal purposes...."
Now, under no circumstances am I saying that this provision gives a school permission to record and publish (post) a copyright-protected musical work. But a copy that is created incident to streaming can be retained by the school or teacher, and perhaps posted to an intranet, if they plan to use it for rehearsal or evaluation later.
The important take-away from all of these is: your school may have options from not only within but additional to the license. By assessing the precise permission your school received, the reasons for recording, and the reasons for posting, a school can consider their full range of options.
Of course, what copyright law can give, contract law can take away. So, if your school has secured a license with a specific agreement that you will not make and post a recording, remember that's a contract term it agreed to, even if fair use would otherwise authorize the use.
I know, I know, thinking about copyright while planning to make the most of a performance can feel like allegro, adagio, adagio, allegro...
Just andante, like the question models, plan what you need, and you'll find a good pace!
Thank you for a thoughtful question.
 I know "streaming" and "recording" are different, but as a technical matter, "streaming" does create a digital copy, even if it is fleeting.
 This answer does not consider limited posting on an intranet, although I'd argue that with planning such posting could be consistent with the CONTU guidelines for retaining a copy for rehearsal or evaluation.
 This is why people negotiating for license content should always be trained to not negotiate away rights your district has by law.