I'm using Buddypress. I just want to add some kind of language tool that syncs (or updates) with the official WordPress website’s translations for plugins
Plugins should come with all their language files already, all it requires is setting the language of your site which is done on install. That will set which language the Admin area uses, etc
You can change this under general settings.
Note that BuddyPress may not have been translated into your language of choice, not all plugins have translations, or are built in a way that allows it to be done. This is usually because the plugin author didn't know how, or it did not occur to them to do so
and everything throughout the website and let’s users choose between languages (like some kind of language switcher dropdown)
Content on the other hand is entirely the domain of plugins. WordPress doesn't have built in multilingual content support
I tried using polylang but it seems to be using a translation of it’s own.
WP can't translate your content for you, Polylang will be doing its own thing, similar to other multilingual plugins. They also work differently, so one plugin may not work the way you expect depending on how you want translation to work. E.g. do you want a french translation of your posts, or a french sub-site? You need to think about these things while choosing a plugin
Plugin recommendations are out of scope on this site. If the only aspect of your question was which plugin to use, it would be closed off topic. Remember, you need to be able to mark an answer as the correct answer, not just what worked for you this time