I've just completed my test.
I've launched the vSphere Client (flash) and the tag category was renamed... I logged again in the HTML 5 client, and the tag category did show the new name.
So I retested a rename in both client: in flash, the change is immediate, but in the HTML 5, even logoff/login or pressing F5 did not update the change.
I change from Firefox 69 to Chrome 77 and the tag category was named as in Firefox 69 flash client...
It seems like the HTML 5 client is lazy to update the interface to reflect the changes...