My understanding is that the notification “Item saved” is initiated by the server upon successful update of the cloud database; there are alternative messages that indicate errors in saving the updated vault item.
If you have evidence that the Web Vault is not updated after the “Item saved” message has been displayed, I would like to see it (or at least have get your confirmation that you have observed such behavior).
I think it would be challenging to poll each logged-in device to verify that the cloud database update has propagated to all Bitwarden client instances before displaying the success message, and attempting to do so would certainly introduce a noticeable delay in the display of the message.
For the record, the Bitwarden code is not my work. I am a Bitwarden customer, same as you.