Earlier quoted context omitted.
How does the language cookie get set in the first place? Is there any option for the user to set his preferred language?
It is a part of browser's settings (for Chrome and Firefox at least), and it is default to system's default language I think.
To answer the question, yes, normally you would have some kind of manual switcher that sets the cookie. Please don’t use flags for this, because flags indicate countries, not languages. Don’t make a Russian-speaking Ukrainian seek out the Russian flag, and don’t make an English-speaking Irish person seek out the Union Flag.