Nietzsche often uses the same word to mean different things. When he uses the word "nihilist", he could be referencing three separate ideas:
1) Christian Nihilism - the negation of life and movement toward nothingness; Christianity does not feel itself to be nihilistic, but from the atheistic perspective Nietzsche has, it aims toward nothing and denies life. While the atheist has only life to value, the Christian decides that this life we live now has no value, and only the life beyond has meaning. However, since that life is non-existent, Christians value nothing at all.
2) Negative Nihilism - the realization that the universe is a cold and unloving place, and thus the belief that one's life has no meaning. This is the kind of nihilism the OP is talking about, and it is a nihilism that Nietzsche is fighting against because he believes that it is a natural post-Christian tendency.
3) Positive Nihilism - the realization that the universe is a cold and unloving place, and that is perfectly fine. While life has no objective meaning, the positive nihilist has come to realize that the valuation of the universe takes place solely within his/her own mind, and he/she can choose what meaning life has. This is could also be called a "happy nihilism", because the burden of God has been destroyed, but the shadow of God, negative nihilism, has been destroyed as well. What is left is a love of life.
I think the main reason people struggle to read Nietzsche is that they do not catch these subtleties; Nietzsche does us no favors, and uses the terms interchangeably. His goal is not necessarily to help us understand him, but rather to create introspection.
First, let me say that I like that Nietzsche is being discussed. He is often overlooked (in name) despite being the most influential philosopher of modern times. No one is better at kicking these often trite discussions in the balls.
What you present is a little confusing when trying to see the larger picture Nietzsche painted and I think becomes a little misguided towards the end. Rather than framing everything in terms of nihilism, we should consider more Nietzschean concepts: Death of God, Life-Affirmation vs. Life-Negation, Master/Slave Morality, The Last Man, Herd Mentality, and The Will to Power.
I actually was going to write a mid to larger size paragraphs for each of these, giving a brief run down of the idea and how I would apply it to this conversation, but I just got too damn tired since my lack of sleep is catching up to me and this is pretty energy consuming. I don't think you can just wikipedia this stuff to really grasp it (although, I wonder if me providing some simple summaries is any better), but if anyone reading this is new to the man, it may not hurt to at least google the terms above.
Anyway, so that this post can have some substance...
We shouldn't overlook why Christians (or similar) resort to life-negation and in what ways they do it. Heaven is the ultimate conclusion of their life negation (trading a bad life for an infinitely good one that follows it), but there are also the virtues (inverted values) borne from ressentiment (jealousy, weakness). This is useful, because we can see the very same negation in almost all modern ideologies/moralities. Many posts in this thread depend on such an ideology. Appeals to compassion or community, such as "You need God to be moral?" are uncritical answers, either simplistic or ignorant acceptance of one's morality - namely, in this case, humanism, which Nietzsche himself foresaw as an secular extension of Christianity, no less influenced by slave morality and herd instinct, and championed by the Last Man. Nietzsche saw the death of God as nothing to be trivialized, thus sought to find the origins of morals (a genealogy). His writings on why people come to accept moral facts is particularly relevant to this discussion even if we ignore the belief of God.
The flip side to that is life-affirmation. This is mostly what you mean with "Positive Nihilism", although it comes across as a misunderstanding and doesn't quite do it justice. Allow me to recycle some of what I would have written in the previous version of this post:
Life-affirmation can boiled down to the phrase "amor fati", the love of (your) fate. Since people naturally want to say to themselves "I want to affirm my life!" they find the idea appealing in a self-help book sort of way, but it takes a lot of strength to truly affirm life and people only see the idea superficially. It's not about being happy with yourself; it's about accepting that people get robbed, raped, and murdered (including even yourself), people are born to die painful and pointless deaths, people will treat you or others "unfairly" based on race, sex, etc. (indeed, the very idea of "unfairness" is opposed to that of amor fati); it's embracing your own suffering (not trying to mitigate it like with Buddhism or Schopenhauer) and embracing the suffering you will cause; finally, it's facing the tragic meaningless of life (which means accepting that it is indeed tragic, not simply something you can trivialize) and poetically finding meaning in it anyway. It's common enough to try to simulate this through pure hedonism ("Will to Pleasure"), one possible reaction to nihilism, but Nietzsche would say it is not the real thing. Likewise, he would be particularly critical of those who simply continued Christian values while thinking they had affirmed life simply because they refused God, acting like droids following their programming long after their human masters had perished. In any case, Nietzsche did not believe he himself was strong enough to truly affirm it in its entirely; that would be the role of the Overman.
Nietzsche wouldn't be too comfortable with the idea of just looking at the "mind". He believed values had their basis in instincts. The physical/mental distinction is weak here, thus, you only get to "decide" your values to a point, e.g., if you are ugly on the outside, you will also be ugly on the inside. Weak and oppressed people develop reactive (slave) moralities in relation to the strong and oppressive people.
I think the idea of "the burden of God has been destroyed" is the one part I must disagree with entirely. Nietzsche goes to great lengths to express how such a burden (or such a crime, as "we killed him") has in no way been lifted. He instead chooses to replace it with the ideal of going from man to Overman, a journey similar to the one we made going from animal to man. You can say he is trying to find the God "inside" us. And the "shadow" is not to something to be dismissed, but rather something we must overcome (overcoming is everything for Nietzsche, really). You call it "Positive Nihilism", and the way you phrase it reflects this, but Nietzsche really wanted to defeat nihilism. A love of life is not something you gain by removing burdens (i.e., more negation, very Buddhist), but, rather, something you can't help but express to yourself as you stare tragedy in the face and smile, in an herculean feat.