PRO's.... These are the folks who collect the performance royalties for the public performance of the song on TV, radio, concerts, and other live events. You can only join one. Since they all basically do the same thing, you join only one and stick with them. they have slight differences but the payment formula tends to work out pretty close to the same no matter which one you choose.
Beagle..... all songs when written as you know are copyrighted by common law to the writer when they are recorded in a tangible way. If you self publish..... create your own publishing company and do all the footwork to get the songs to the artists, you can retain the copyright in your name.
It is common practice to sign the song to a publisher. And when you do, you are signing it away. You give up ALL rights to it. So if you wanted to do a cover of your own song..... the publisher would control that right and you would ask them for permission. Sounds like a bad deal but it's really not. In the normal contract, you retain what is called writer's share of all profits (net) and the rights to being recognized as the writer of the song..... "writer credits" ... in all uses of the song.
The publisher, generally has the contacts and connections needed to get the song to producers and artists that you as an unknown (or known) writer can very often NOT get to. They do the hard work of plugging the song and getting it "published" in a commercial sense. They retain the "publishers share" of the profits (net) earned from the exploitation of the song.
In the example you gave you are correct. HFA administers the licenses for use.... they are a specialty company which fills a need. Yes.... Bob Segar wrote it and still has writing credits..... but at some point he assigned his ownership rights to a publishing company which now owns the song. Bob may own that publisher or he may not.... either way, of 100% of the earnings minus fair and acceptable overhead costs, the net profit gets split 50/50 between Bob and the publisher.
All the big artists work this way. Check any CD by a major artist. Every song is indicated as copyright to a publisher. I'm sure there are exceptions but this is general practice. Artists who have released several CD's and have a few hits generally will at some point start up their own publishing company to capture that other 50% publisher share. Even some prolific hit writing songwriters will start a publishing company once they get a few hits and get to know enough people to make it a viable solution.
So... I would rather get 50% of the royalties of a song that I signed to a publisher that to retain 100% of a song making nothing.
A recording contract is NOT the same... not even close, to a song publishing contract. Most song/publishing contracts tend to be individual contracts..... meaning one contract for one song.
Some publishers hire and sign writers to a songwriting contract...again, this is a variation of this publishing thing. Staff writers, as they are called, have the advantage of being in a building with other writers and can often co-write with big name writers, but they don't have near the writer share percentages in the songs as they would like.
Its a scary thing signing away the ownership of your songs to a publisher. It certainly pays to do your homework and know something about the company before you sign on the line. But that's the most efficient way to get your songs out to the artists unless you happen to be in a band with someone like Allan Jackson, Carrie Underwood, or Reba.