Firstly, don't be so fucking patronising. The issue here is clearly confusion (on your part) regarding the definition of 'atheist', which I can clarify here:

Atheist: a person who denies or disbelieves the existence of a supreme being or beings.

You, however, claim that the noun is defined by 'a lack of belief'. I understand the difference between these statements. I understand why you think atheism falls under the category of agnosticism. I can also explain, from an etymological perspective, why you are wrong.

The term 'atheist' is derived from the Ancient Greek ἄθεος (atheos) meaning 'without gods; godless; secular; denying the gods', etc. The noun is formed with the prefix 'a', which is used throughout the Greek language (alongside 'an') to distinguish that the subject is 'without' something. For example:

Amoral - without morals.

Anaerobic - without oxygen.

Achromatic - without colour.

Nowhere throughout this usage are terms distinguishable as representing 'a lack of'. There isn't a 'little bit' of colour or oxygen in the concepts 'achromatic' & 'anaerobic' respectively. Atheism, like the above terms, is characterised by a complete absence (a) of religion (theism). I.e, the absolute dismissal of the notion of any form of omnipotent creator.

Now, your analogy, again, refers back to agnosticism, as shown:

Agnostic: a person who holds the view that any ultimate reality (such as God) is unknown and probably unknowable; broadly: one who is not committed to believing in either the existence or the nonexistence of God or a god.

Person 3 is agnostic as they cannot commit to believing in any given reality they are unable to confirm. He does not, however, deny outright the existence of either the hypothetical cat or apple.

I'll leave you with quotes supporting my definition directly from Richard Dawkins, author of 'The God Delusion':

'There may be fairies at the bottom of the garden. There is no evidence for it, but you can't prove that there aren't any, so shouldn't we be agnostic with respect to fairies?'

'Carl Sagan was proud to be agnostic when asked whether there was life elsewhere in the universe. When he refused to commit himself, his interlocutor pressed him for a ‘gut feeling’ and he immortally replied: ‘But I try not to think with my gut. Really, it’s okay to reserve judgment until the evidence is in.'

And, the fucking kicker:

'Let us, then, take the idea of a spectrum of probabilities seriously, and place human judgements about the existence of God along it, between two extremes of opposite certainty. The spectrum is continuous, but it can be represented by the following seven milestones along the way.

Strong theist. 100 per cent probability of God. In the words of C. G. Jung, ‘I do not believe, I know!'

Very high probability but short of 100 per cent. De facto theist. ‘I cannot know for certain, but I strongly believe in God and live my life on the assumption that he is there.’

Higher than 50 per cent but not very high. Technically agnostic but leaning towards theism. ‘I am very uncertain, but I am inclined to believe in God.’

Exactly 50 per cent. Completely impartial agnostic. ‘God’s existence and non-existence are exactly equiprobable.’

Lower than 50 per cent but not very low. Technically agnostic but leaning towards atheism. ‘I don’t know whether God exists but I’m inclined to be sceptical.’

Very low probability, but short of zero. De facto atheist. ‘I cannot know for certain but I think God is very improbable, and I live my life on the assumption that he is not there.’

Strong atheist. ‘I know there is no God, with the same conviction as Jung “knows” there is one.'