A professional musician has a similar problem - we must make these kinds of distinctions all of the time. If I am listening to a recording of myself in particular, it becomes nearly impossible to keep these two kinds of listening separate. The term most of us use for the first kind of session you describe would be "criticial" listening. For those of us who are always performing, we must learn to do both kinds of listening simultaneously, but this is no easy task. Generally speaking, the first time I hear something, it is almost impossible not to listen critically. But it is certainly possible, and even essential to maintain sanity to be able to turn off the critical listening and just enjoy the music. Those of us who play full-time in symphony orchestras all have certain pieces that we find very hard to enjoy listening to because we have to play them so often at work - the Nutcracker springs to mind for me immediately, the Pachelbel Canon would be another one for many, as would the Messiah. What I do to try to keep these pieces of music somewhat fresh is never listen to them outside of work. You certainly won't find a recording of the Nutcracker in my record collection.
That said, I always encourage non-musicians (I hate that term, actually) to develop their ears, as this kind of understanding will always enhance their enjoyment of all types of music, and this definitely involves critical listening. So it is not necessarily desirable to always keep both kinds of sessions separate. The more you listen critically, the more you will learn about the music you are listening to, and the more enjoyment you will receive from it when not listening critically. I have trained many people's ears, and not once has any of them come back to me and complained that they didn't enjoy listening as much anymore. To be more specific for audiophiles, I would say that doing more critical listening in general will also increase your ability to determine what kinds of traits you like in the equipment you are auditioning. One shouldn't become obsessive about it and forget to enjoy the music at all, but it is great fun to listen to the same recording on different pieces of equipment, or different pressings of the same recording on the same piece of equipment, and many of us who are professionals do this all the time, both for study and for pleasure. The more you know about what is going on, the more things you have to take enjoyment from.
That said, I always encourage non-musicians (I hate that term, actually) to develop their ears, as this kind of understanding will always enhance their enjoyment of all types of music, and this definitely involves critical listening. So it is not necessarily desirable to always keep both kinds of sessions separate. The more you listen critically, the more you will learn about the music you are listening to, and the more enjoyment you will receive from it when not listening critically. I have trained many people's ears, and not once has any of them come back to me and complained that they didn't enjoy listening as much anymore. To be more specific for audiophiles, I would say that doing more critical listening in general will also increase your ability to determine what kinds of traits you like in the equipment you are auditioning. One shouldn't become obsessive about it and forget to enjoy the music at all, but it is great fun to listen to the same recording on different pieces of equipment, or different pressings of the same recording on the same piece of equipment, and many of us who are professionals do this all the time, both for study and for pleasure. The more you know about what is going on, the more things you have to take enjoyment from.