Re-appropriating the term atheism is what is causing the issue. If we break the word down it is clear that the first definition is what should be intended when using the word atheism. The prefix a has long been used as a reversal, or negation, for the term that follows it. The suffix ism has long been used for any kind of belief or organized thought system. The actual term is derived from the Greek word for god.
Here's what we are left with:
- Theism is a belief in god.
- Atheism is a belief in not god.
The confusion leading to this re-appropriation is from misplacing the negation. If we were to say that atheism means not a belief in god (notice the misplacement of the reversal/negation) then we are not being true to the word's structure and implied definition. The ism suffix already implies what it is, a belief or school of thought, and placing the negation on that is contradictory. The a does not negate the ism; it negates the root, theo. It is akin to saying that it is not a belief or school of thought, but (and the following might be better on philosophy.se) that is the same as saying that atheism and theism are in fact not opposites and are therefore in a different class of terms, which can lead one to think that they are not mutually exclusive, however their regular use suggests otherwise.
Atheists who want to re-appropriate the term's meaning are insistent that what they hold to is not a belief (however they might likely accept school of thought), so to overcome the above contradiction they replace the negation with the word lack, leading us to the second definition that you provided.
There is already a word for that, however: agnosticism. Agnosticism (also a school of thought, but not necessarily a belief) is characterized by confessing uncertainty on the topic entirely. Essentially, saying one is agnostic is the same as saying that one is unsure on whether there is a god or not. They are unsure if belief in a god is warranted, therefore, there is a lack of belief.
If one would characterize his opinion on the topic as certainty that there is no god (notice the negation placement) then one should rightly call oneself atheist. If one would characterize his opinion on the topic as not having a belief or opinion (notice the negation again) or a lack of belief or opinion then one should rightly call oneself agnostic.
Now this is a sensitive topic for some people. Trying to place what you believe into a single word is difficult at best and laying down the three terms, theism, atheism, and agnosticism, and demanding that all persons fit neatly into only one of them is not fair to them and does a great disservice to the greatest of all human quests: finding meaning in our lives.
But with that said, the re-appropriation of atheism is confusing at best and completely redefining it at worst. It would be best to stick to the original meaning, the first definition that you supplied, and if one finds oneself not quite fitting into the mold of one of the three then err on agnosticism when forced to use a single word and speak up to be heard when there is no such restriction. Even coin your one phrase if necessary; it might catch on.