All right!
I removed the code and made a full update.
Patience was again the key after the update. Translations were not working, layouts were broken at first. Clearing the Cornerstone System cache didn’t helped immediately, but resaving permalinks did. The site gradually fixed itself after the update.
WPML has integrated what used to be the Translation Management addon into the main plugin. Now if I click a page to edit it, it offers all the strings side-by-side.
I have no intention of using this method since we have the Cornerstone flags that we can click and edit the translations that way. One thing that is interesting is that, when the page is translated by switching Cornerstone flags, the Translation Management doesn’t see any translations. It thinks that the page is not translated and it offers an option to start translating.
This is a bit conflicting and potentially confusing for users. I guess the rule would be “Just don’t touch the Translation management part when working with Cornerstone pages”.
On the other hand, when translating stuff such as the Cart page, then the Translation management is the right way to do it.
I don’t even know what is my question here. Just an observation, lol. 
Thanks!