Shardone, that's a good definition of a producer, but it doesn't fit all situations. A current trend in pop music is to bring in a different producer for individual album tracks. For a flat fee the "producer" comes in and sprinkles his signature sound upon the artist with the goal being a hit record. In other situations there is no producer. A band without a label rents a few days in a studio, hires an engineer and record enough songs for a CD. If the band has no studio savy, the engineer will have to do things a producer will normally have input on, but essentially the CD has no producer.
It's more likely that a producer will have engineering chops than an engineer having a true flair for production, but the line is still very fluid with people doing both jobs. It's common for an engineer to ask for production credit, but rarely the other way around.
