Typically, the latter is pursued by recording companies, because that is the IP they own or exclusively license. The rest is rarely handled by the individual rights-holders, as they usually just let ASCAP or a similar organization handle the business and litigation end, and just cash the royalty checks.

Licensing the lyrics should be a whole lot cheaper than the music or a specific recording. So it would not cost a lyrics site as much to be fully legit as it would for a music streaming site. Likewise, it may cost less to license a cover band to perform something than to license the recording made of it by a famous group, if the song was not actually written by them. This is why ABC's Dancing With the Stars uses so many covers, but obviously still uses recognizable recordings where Disney already has or can cheaply obtain a performance license.