Job of Producer versus Recording Engineer

producer is the one who financially promotes albums and artists.

recording engineer simply the one in-charge of the recording process of the above mentioned items.

I'm not sure that that is a correct explanation with respect to recording as opposed to movie/theater. My understanding is that the producer of a recording directs the mix, sets the flavor, chooses the overall direction of the recording. Well known producers have a "sound" that they are known for, like Phil Spector, Don Was, Quincy Jones, etc. I may very well be wrong, but I thought that a record prdocuer is more like a director of a movie or play. The money is usually put up by the record label.

The recording engineer will set up the mics and associated gear,and pending on the type of producer, some more involved in the actural mix then others,may do the mix as well. But the final tracks at least would be given the ok by the performer and producer.

It's not always clear cut, but the producer is responsible for the overall sound and vibe of a recording while the engineer is responsible for getting that sound on tape. It really depends on the specific people involved and the roles can heavily overlap.