Just use OsmAnd. It’s old-school in terms of its UI and can be clunky, but it’s FOSS, has hourly map updates (free for OSM contributors), and a lot of customization and features.

It’s kind of a malicious compliance thing. The new name will not show up on any map renders, moreover it’s not likely to be shown in most apps even if you select the object (because official_name:* is not supported by most apps).

The fact it doesn’t show you the best match of your search based on distance by your location (or is because)

That highly depends on the search engine you’re using, OsmAnd for example sorts the search results by distance to your location.

It doesn’t show the 2 places I sent ~1 month ago or they’re hard to find (available on site, not on organic maps)

That’s an Organic Maps problem, they only update the OSM database once a month, and you have to click the update button manually when they do so.

I kinda recommend against Organic Maps at this point. It’s a dumbed-down app with bad routing, bad search, and slow updates. The only thing it has going for it is that it has the best UI/UX (especially for new users) compared to other FOSS openstreetmap apps. If you’re ok with proprietary nonsense, mapy.cz is a lot more fleshed out, and otherwise you should just learn and configure OsmAnd for yourself.