Wikipedia's articles reflect a very controversial and difficult topic, which is probably why they are so unintelligible. Let me make this clear right at the start of this attempt to answer your question - it is very difficult to find a communis opinio even within the relatively small scientific community that works with ancient gnostic texts on what "Gnosticism" or "Gnosis" actually is and which texts really are "gnostic" and which aren't. You could probably write a whole book on different definitions of "Gnosticism" and/or "Gnosis", just for the ancient texts, not even including the later forms of "Gnosticism" (or "Gnosis") you briefly mentioned.

So let's start at the very bottom - "Gnosis". What does that term mean and why do we use it? "Gnosis" is a Greek term, as I am sure you already know - Gnosis (γνωσις) means something like "knowledge", but in the sense of a perception, an insight you get, not something you just have. Nowadays "gnosis" as a word has a somewhat esoteric feel to it, from the Christian perspective it's of course utterly negative, but that wasn't always the case. As an adjective, "gnostic" (γνωστικος), was used to describe someone as well-informed, knowledgable. In fact it's used in that way in many Early Christian sources (e.g. Barnabas, 1 Clement). Even Irenaeus, the great Christian apologetic, in his books Adversus Haereses (which were targeted at "gnostic" sects) was careful to explicitly talk about "false Gnosis" when combatting these religious movements. The terminological innovation in Irenaeus' books is that he describes members of those sects as "Gnostics" (γνωστικοι), which led Henry More (1614-87) to "create" the term "gnosticism" as a (pejorative) umbrella term for the movements described in Irenaeus' books. Clemens of Alexandria uses "Gnostics", too, but he distinguishes "true" Gnostics from "false" ones - "true" Gnostics being those who are "perfected" Christians, those who reached a higher form of spirituality through some form of "Gnosis"; "false" Gnostics of course being those who follow wrong teachings and whose "Gnosis" is nothing else but delusion.

We have no actual "gnostic" sources that would allow us to say with certainty that these sects called themselves "Gnostics" at any point in their history. The terms we use are alien creations, that imply sort of a common ground between the many sects ancient authors like Irenaeus described. But surely the ancient Christian apologetic must have seen some sort of "common ground" indeed, if he writes books about false "Gnosis" and is able to name many sects that propagated such false "Gnosis" - and the many modern authors, academics as well as laymen, they have to see that "common ground" when writing the myriad of books called something like "Secret Gnosis", "Early Gnosticism" or "Mysterious Gnostics". That is very true indeed, but the difficulty here is that there is no widely accepted definition of "Gnosticism" or "Gnosis". Every modern book on "Gnosis" has to establish for itself what it means by "Gnosis" or "Gnosticism", because - as I wrote in the very first sentences - there is no communis opinio left on that matter.

Many nowadays refer to the conference of Messina in 1966 where scholars tried to establish a definite definition for both the terms "Gnosis" and "Gnosticism". According to the wise men gathered there "Gnosis" is supposed to describe a general form of knowledge, acquired through some form of divine revelation, but whose understanding is limited to a small elite which keeps this knowledge to itself - so "Gnosis" could be used to describe a myriad of sects with some form of "secret knowledge", not just the ancient ones. "Gnosticism" however is - according to the conference - to be used as a narrower term to describe a specific religious movement of antiquity. The conference did not fail to give us a definition on what "Gnosticism" is supposed to describe. It's "a dualistic conception on a monistic background, expressed in a double movement of devolution and reintegration". This "double movement" is "the idea of a divine spark in man, deriving from the divine realm, fallen into this world of fate, birth and death, and needing to be awakened by the divine counterpart of the self in order to be finally re-integrated" and "conceptions of a 'devolution' of the divine, this [...] idea based ontologically on the conception of a downward movement of the divine whose periphery had to submit to the fate of entering into a crisis and producing - even if only indirectly - this world, upon which it cannot turn its back [...]." These are very fancy words - let's try and dissemble them.

The whole idea of the cosmos and the man (as a species) within the cosmos is based on the idea of this "double movement" - it's both devolution, "going down", and reintegration, "going up". Picture this on a mental image of the cosmos, the bottom is "matter", everything existent like you yourself, your house, your garden, your cat - the elements and so on; the ceiling is "heaven", a far away sphere that you - as a being of matter, made of weak flesh and bones - can never understand, it's the ultimate "good", eternal light and so on. But despair not, little fleshling, as there is hope! This derives from a popular understanding of platonic philosophy - everything that we see is based on a concept, on an "idea", that exists in an otherworldly sphere of ideas. So, in a simple example, every table you see is different but yet you can easily distinguish a table from a chair - that's because every table you see is nothing but an imperfect image of the "idea" of table, lingering in the sphere of the ideas. It's the same with man - every man is different, yet we can easily say who's a human and who is a mouse. This is the "divine man", the first image of man, from which we all are just imperfect images ourselves, the "blueprint" of our human life. For some reason - tragic or evil in most of the "gnostic" texts - man was created after the image of the divine man, and in that process a "divine spark" of him, of this otherworldly sphere, the ultimate good, was captured within our fleshly bodies. This is often beautifully explained in long and very elaborate myths that can differ quite a lot from each other. The "fall" of this divine spark, to be captured in our bodies, that is the downward movement, the "devolution", as the conference explained it. Hope with the promise of final salvation comes from the upward movement, the "reintegration". This "divine spark" from the upper sphere lingers within us, but very few are actually aware of it. The idea of this "divine spark", too, stems from popular platonic philosophy, which was all the rage back in those days - as humans we are tripartite, we have a body (σωμα) and a soul (ψυχη) and mind (νους); now, animals only have a body, which you can easily see, and a soul, which makes them move, which makes them eat and which makes them copulate - in short, the soul is the necessities of our fleshly life; but - in contrast to animals - we have the "mind", a higher form of intellect which allows us a deeper form of understanding, which allows us to speak, to plan ahead, to perceive something as "divine" and so on. Now mind is the "divine spark" that was given to us from the otherworldly sphere, but since most of us aren't aware of our mind (little has changed here, one might be tempted to say), we need to be "reminded" (haha) of it. This happens through some form of revelation from the divine sphere - in Christian Gnostic sects often through Jesus Christ himself. "Gnosis", knowledge, here is the ultimate tool to become aware of yourself, and when you're aware of yourself - meaning you clearly see yourself as a "divine" being, ultimately foreign to this sphere of matter - you will rise up through the spheres of heaven back to the divine man and you will become one with him and stay where you belong - at the side of the ultimate good, that is god. This god is very foreign to us, we cannot understand him, we cannot describe him, we shouldn't even try - yet we have to recognize the divine within ourselves and therefore god himself to reach for our own ascension. The good forces (below god, for god doesn't move, doesn't act but acts through the good, because he is in everything and everywhere, he's the ultimate calm and peace) have an interest in helping us returning, because the divine spark doesn't belong in the material sphere - we don't belong here.

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