I'm not sure where you got spaghetti westerns from, but the examples you've given in your question are generally considered to be on the pop side of things.

This dichotomy of "pop vs. classical" or "art music vs. pop music" and the question of what defines those things is truly a complicated one in my mind; in that nearly any definition you can come up with is going to cause disagreement, and many hold that the dichotomy is a false one to begin with.

Nevertheless, there are some distinct and undeniable cultural strokes out there, and the terms continue to be useful despite some artists trying their absolute hardest to obliterate the dividing line, or at least what it means to them.

User filzilla has already posted an excellent overview of some of the most important music written in the past few decades, so rather than rehash or add to that, I'm going to try to speak to the definition side of the question in some more concrete terms and attempt to provide some backstory.

Many people are aware that a tradition exists of "classical" music; the instrumentalists and singers who go to conservatory to study and build their skills so they can sing at the opera or play in an orchestra all of the well-worn chestnuts by Bach, Beethoven, Brahms, and the rest of them. What fewer people are aware of, however, is that there is a stylistic lineage in this tradition that has stretched unbroken across time from these household-name composers of old to the present day. Now, this lineage was a fairly tightly-wound braid up until the middle of the 20th century, when some key styles started to diverge from one another before everything shattered into a thousand pieces in the 1970s (the "stellar nursery" mentioned in filzilla's answer). In other words, it used to be relatively easy to track what the style of present-day music was and where it was headed, but these days no composer really feels beholden to follow in the stylistic footsteps of their teacher, as was previously the norm. The lineage, though now fragmented, it still there; but now new influences are being integrated faster and in more places than we can really keep track of for art music on the whole.

In light of this, here are some of the (hopefully) least contentious "traditions" that still serve to differentiate "art music" from "pop music", understanding that there are always exceptions:

• Art music is written so that a piece or work makes an artistic statement and is seen as the artistic product, not the album, as tends to be the case with pop music. There are no "singles" in art music.
• Art music tends to have a division of labor present between the composer and performer. As such, performers must be trained to the highest possible level of musicianship and technique so they are capable of playing whatever is thrown at them. Composer-performers exist, certainly, both in the past and present, but do tend to have the same kind of training.
• Also in the division of labor vein, composers tend to be supported monetarily by commissions, royalties, grants, and sale or rental of performance materials (sheet music), while performers tend to be supported by recording album contracts and the performances themselves. Contrast with pop music, where one artist (often aided by a producer) writes, performs, records, and tours all at once.
• If you're at the Grammys or something else that calls itself the "music industry", it's probably pop music. If the MacArthur Foundation or someone else is giving you free money just so you can keep making music (without taking a cut of your royalties), it's probably art music.
• With the obvious exception of opera, a performance of art music is generally focused almost entirely upon the music itself. There is no mood lighting, the stage is bare without distractions, the performers are dressed simply and/or elegantly (usually to provide no distractions). Pop music tends to be focused upon five-sensory spectacular impact. Music videos! Costumes! Props! A dress made of meat! Synchronized hip gyration!

Of course, immediately upon coming up with some potential criteria, one starts thinking of exceptions, and I truly mean that for all of the items above. The lines are continually being blurred between concert music, opera, and performance art, people are making pop music that's artistically fascinating and art music that sounds like pop music, and it's impossible to pose any definition without someone writing a very long blog post about why your definition is wrong!

I recently spent an entire semester in a class, the aim of which was to pose this very question; and while much excellent discussion was had, we never really came up with a truly reliable answer (and that was the point). At the end of the day, the musicians and composers and audience members that create and appreciate music out there tend to classify themselves and who they musically associate with on their own, and those who appreciate "both" kinds of music either create their own definitions, classify on a case-by-case basis, or refuse the dichotomy outright.

But of course, the term "contemporary art music" still has enough meaning that the people on this site immediately know exactly what you're talking about, so there must be something, right?