Billboard has many different charts, including charts for overall popularity and broken down by various genres, like the Hot 100, the Billboard 200, Pop Songs, Adult Contemporary, Hot Country Songs, and Hot Rock Songs. How do they determine which songs belong on which charts?
From what I could find; in the end it comes down to a chart manager. The Billboard wiki page mentions the following: "What separates the charts is which stations and stores are used; each musical genre has a core audience or retail group. Each genre's department at Billboard is headed up by a chart manager, who makes these determinations."
The charts are based on "key fan interactions with music, including album sales and downloads, track downloads, radio airplay and touring as well as streaming and social interactions on Facebook, Twitter, Vevo, Youtube, Spotify and other popular online destinations for music" (reference).
An overview of specific charts criteria can be found in the chart FAQ. E.g., here is an example of the Trending 140 FAQ, which a.o. compares chart methodologies. Also, the aforementioned wiki page provides an overview of specific chart criteria (see the column 'Description' in the table, starting here). E.g. 'Adult Contemporary has the criterium "Measures airplay spins on 85 adult contemporary stations". So, from what I understand, if you're on that list of 85 stations depends on the decision of the chart manager.