Hi Herman, and Welcome!
I am on the right track to distinguish long term music listening with recording sound studio?
There are differences in layout, yes, but the basic principles are still the same. Both are what is technically known as "critical listening rooms", where the goal is to have the room sounding as neutral as possible, in terms of both frequency response and also in time-domain response. The idea is that the room should neither add anything of its own nor subtract anything from the natural direct sound of the speakers.
Accomplishing that isn't just a matter of putting some foam on a wall: it's a whole series of things, starting with the shape and dimensions of the room: some combinations of room dimensions are beneficial, and help you towards the goal, while others are adverse, and fight against you. This concept is known as the "room ratio", so you need to start out by looking at that: How wide, long and high is your room? That is normally expressed as something like "1 : 1.19 : 1.7 ", where the first figure of "1" represents the height, the second tells you how much wider the room is than high, and the third tells you how much longer it is than high. Over th years, scientists with names like Louden, Bolt, and Sepmeyer have figure out what the most favorable ratios are, and also what the worst ones are, so you should start out by making sure your room is close to one of the "good" ones, and far from all of the "bad" ones. If not, then you will have to adjust the length and/or width.
Once you have a good ratio, the next issue is to get your speakers and listening position in the correct positions, and for what you need, the room will have to be laid out more like a Home Theater than a Control Room: listening position towards the back wall, not towards the front wall. There's a couple of ways of doing that, depending on how far you want to go, and how much you want to spend, but the basic concept here is to provide even coverage of the listening are while maintaining stereo image and sound stage. It sounds a lot easier than it is!
Then, with the speakers in place, only then can you place the treatment, and that too depends on the room shape/size, and the relative locations of the speakers and listening position. When it comes time to design the room treatment, it is also a good idea to run an acoustic analysis of the room, using software called "REW" (which is free!), and posting the results here, so we can see how the room is behaving and suggest ways you can fix it.
So there's a process here, that starts out with the dimensions of your room. If you can give us those and also post some photos of the room, then we can give you the basic pointers. We also need to know exactly which speakers you plan to use (make and model), and any other details of the way you plan to furnish the room (chairs, etc.).
Question: is there if any, specific approach to treat the room for natural, pleasant sound for long term listening sessions, for generally broad listening area (say 10 to 20 people in 500sq ft) where priority is natural sound?
Yep!

See above.
Question 2: Is there a way to make the wall behind the speakers appear acoustically pushed further back in a NATURAL way, so there is a deeper sound stage? I tried installing 2inch sound absorption panels on this wall and it sounded very strange and unnatural as if there is an energy hole between speakers. not pleasant.
It's not just the front wall that is involved, but rather the entire room. However, yes, there is a way to take the front wall out of the equation, and that is known as "flush mount" or "soffit mount", which basically means that you mount the speakers inside the walls, such that the front panel of the speaker (it's "baffle") ends up flush with the wall: in essence the speaker is no longer "inside" the room, but is now part of the wall, so all of the artifacts caused by having speakers close to walls (SBIR, comb filtering, phasing, and all those other things) are simply gone. They just don't exist any more.
It's a pretty drastic move to do that, but it works wonders for the acoustic quality! As you found out, 2 inches of absorption by itself doesn't do much...

It takes far more drastic steps than that to make a room sound good.
So that's the basic question: how far are you prepared to go to accomplish your goal? If you say, for example, that you want to spend no more than US$ 500, then you are pretty much out of luck: that won't accomplish much. You an do some very basic things for that, and it will certainly improve, but it won't be excellent. At the other extreme, if you say you have US$ 50,000 to spend, then you can get it sounding pretty darn amazing! Like most things in life, unfortunately, the final outcome depends greatly on how much you are prepared to invest in getting there.
So if you can also let us know your budget, that would help to put things in perspective. There's no point in talking about soffit mounts and slot walls if you only have US$ 500 to spend, but there's also no point in taking about cheap foam and minimal thickness if you planned on spending US$ 10,000 to get it right...
- Stuart -