Wrong language in a new Chrome private window and in integrated data
I actually solved my problem myself. On the Chrome app, I clicked on Settings-> Languages (in the Advanced category) and then I removed French from the language list, which apparently set to be my first language, I don't know for what reason. Now pages do not display anymore in French, so the problem is solved.

I would like to keep the answer on the website, it might be useful to someone, but admins can delete it if they think it is not useful.