The Boston Tea Party Ships & Museum reopened in 2012 with the Robinson Tea Chest as one of their main attractions. The site referenced by OP lays out the provenance of the chest, which is largely considered to be an authentic relic of the Boston Tea Party. When the museum opened it was believed that there was a second surviving tea chest and this was reflected in promotional materials and in onsite tour content.
What has been referenced as "the other surviving chest" is a chest at the museum of the Daughters of the American Revolution, which is no longer considered to be a survivor from the Boston Tea Party. It's recently been dated to a few decades after the event. You can see it in their collections.
You can read more about its story in boston1775.blogspot.com
During my time as a supervisor at the Boston Tea Party Ships & Museum (2016–2018) we had correspondence with the DAR museum, wherein they confirmed that the chest in their possession was no longer considered a relic of the Boston Tea Party. We then began training actors to speak of the Robinson Tea Chest as the only surviving tea chest. I can't answer why this page on the website has not yet been updated with this change.