Background of the issue:
I found a bug while editing the slug of a translated page. You can find how to reproduce the issue on the video: hidden link. This is on a fresh install with nothing else than the official classic editor plugin, WPML & string translation plugins. I just set-up 2 languages, created a page and its translation and recorded the video.
Symptoms:
I expected to see the correct permalink for the page I'm currently on. Instead, I got a mix of permalinks between the last updated translation and the current one.
Questions:
Why am I seeing a mix of permalinks between the last updated translation and the current one?
How can I ensure the correct permalink is displayed for the page I'm currently on?
I just watched the video you shared and please note that is not the correct way of updating translation. In fact, this is a matter of understanding and not a WPML issue.
As per your video, clicking "update" button in English tab reloaded the page and set current language to English. So any change made in French tab without reloading it first, was done for the English language.
It is OK to open different language content in separate tabs, BUT, always reload the tab first to confirm the language you're working on.
Since different tabs in same browser, share the same session, a previously opened tab can be mistaken for wrong information visually. So remember to reload the tab to have the up-to-date content.