Hey
The answer to this comes down to a change in meaning of the word producer, more with amateurs abusing a term they misunderstand than a real change of meaning, though there are some legitimate grounds due to the creep in responsibility for some producers.
Traditionally there are two types or levels of producers. One type takes on getting the best out the musicians in the studio, and helping pull the recording into a cohesive sound, pulling in sound engineers and session people, using specific studios etc to get a "sound". The other type does that plus project manages the whole recording process including budget responsibility.
The rise of home studios resulted in a broadening of the remit. Genres that relied heavily on electronics and software, such as electronica genres, dance/club music, hip hop derived genres, often had a lyricist / vocalist and everything else, from music writing to recording, audio engineering, synthesis, effects, samples etc, was handled by one other person, who quite literally "produced the music". This type of producer role opened the doorway for DJs to crossover, broadening their music creation skills from spinning records to more in depth loop building, sampling and synthesis. You find modern pop producers like Mark Ronson have that type of background.
You can hire producers of any kind on a per song basis, though often established artists engage them on a per album basis if only to help the tracks sit well together.
The traditional method was writer creates demo, demo gets pitched to artists, artists hire producers to create a polished commercial version of the song. You can have several artists recording the same song depending on the agreement. The publisher is the one that gets bands and fim production companies involved. Labels and artists are responsible for specific versions of the song, specific recordings. They plug that to radio stations, tv and film companies and more.
The modern industry has changed a little. With the rise of better quality home recording systems and biz aware indie artists, the industry aren't really looking for demo recordings like they once were, they look for finished recordings, finished arrangements. That is fine for singer songwriters who perform their own material, but more of a problem for writers who don't perform, but then the audience is different. Singer songwriters pitch songs to publishers and a package including songs to labels. Non-performing writers, however, really can target bands as much as publishers. The reasoning is simple. Publishers like to know there is a vehicle out there, ready to start earning. By targeting bands first, they produce finished recordings which can be used to shop the song around. In part the most common approach is lead by what genres you write for. Where providing finished recordings to publishers can be helpful is for plugging the song to tv and film production companies advertising companies even music libraries (if appropriate) where the direct association with an artist is NOT necessary.
There's more but I have to go for now. Talk to you soon.