The term "producer" absolutely, positively, does NOT encompass the beat and the mix. Here is what a REAL producer is (Hip hop is a little different). A producer is someone who works with the artist to get the best possible product out of them. The producer oversees the making of a song. The producer DOES NOT necessarily have to be the one who made the beat. The producer often times hires mucisians, beat makers etc to put together the music, and then he/she will work with the artist to get the best final version of the song. Basically overseeing everything and putting their stamp of approval on it.
Now hip hop, is a little different. In hip hop, the producer is generally the beat maker, and the artist does their own vocal production themselves. Unless you are hiring one of the major players (eg. Dr. Dre, Timbaland, Neptunes), the odds are the artist is going to "produce" the song, and often times the beat maker (or Producer, in the case of hip hop), isn't even present during recording. Most cases, the producer (or producers manager), comes to the studio, plays beats, the artist pics some tracks they like (if any), and then record to them at a later date.
Obviously there are exceptions to the rule. My FIRST major label production just came recently, and it was aactually a great experience for me as a producer, as it was completely different than the norm in hip hop. I produced a track for Consequence (Signed by Kanye to GOOD Music/Sony). This is how it went. I invited him to the studio to listen to some tracks, he picked out one he liked, and started writing right away. He actually didn't even write. He listened for about 5 minutes, then went into the booth ala Jay-Z and just started mumbling a bit, and after a while, he had a verse and spit it. He came back at a later date (last week) and spit is second verse. And now we're gonna go back into the studio and re-lay all the vocals now that he's more comfortable spitting them. All the sessions were quiet. Just he and I, and sometimes his manager. We really got to work 1-on-1 to make a record. And I actually got to "produce" in the real sense of the term. Help him work out different lyrics, let him know when its sounding hot and when its not, etc. Actually injecting the track with my flavor, not just letting him run with the beat.
Now getting back to the topic, the producer will NOT mix the record 95% of the time (in professional situations). The producer creates the song with the artist, and then sends the Pro Tools to a mix engineer to mix it. It then gets sent over to a mastering engineer elsewhere to do the final touch ups and then its ready for radio/CD/etc. In low/no budget situations with unsigned artists or indie labels, then yes, maybe the producer is expected to do a half decent rough mix, but in the professional industry, there are professionals who make a living of just mixing. Who would you rather trust your multi-million dollar budget of a product with, the producer who will give you a decent rough mix, or a mix engineer who will make sure it sound the best it possibly can?
OK WOW, I didn't realize how long this was...Anyways, hopefully this didn't bore you all..