How to make the search engine use search api db?
So ok I found out what is my problem. 
I miss the other entity necessary to build to use the search api db: the search page using the search page module or views, instead of the system search page. 

There is a tutorial here to have a more accurate and complete picture of the process: http://webwash.net/tutorials/intro-search-api-part-1-how-create-search-pages