The phrases, 'agnostic theism' and 'agnostic atheism', are puzzling on the surface. But if we dip below the surface, can we give them coherent sense ? Work out what they might reasonably mean ?
'Agnostic theism' might just mean that one is agnostic about theism. Take theism to involve the proposition, 'There is a God'. If one thought that there was slightly more but not decisive evidence for this proposition than against it and so felt it was right, because of this minimally slight edge of 'for' over 'against', to suspend judgement on the question whether there is a God, then one would be one's position would be one of agnostic theism.
Just the same evidential situation might hold good with regard to atheism. Take atheism to involve the proposition, 'There is no God'. By parity of reasoning, if one thought that there was slightly more evidence for this proposition than against it and so felt it was right, because of this minimally slight edge of 'for' over 'against' or even to suspend judgement on the question whether there is no God, then one's position would be one of agnostic atheism.
The shadow of etymology
I had better deal in advance with a matter that will unavoidably come up : the role of 'a' as a prefix in classical Greek. As a prefix, 'a' can mean 'without' or 'lacking'. On that reading an atheist (a-theist) would be someone who does not have the belief that there is a God. This might be because s/he has never come across the word or the concept. A one-year-old child could be an atheist in this sense, as could an adult into whose mind the concept of God had never entered.
I have no quarrel with this reading of 'atheist' and correspondingly of 'atheism' but I do not believe that it is accurate to the sense of these terms in present-day philosophy of religion. The Greek prefix, 'a', can also carry the sense of denial, or opposite meaning, of the word to which it is attached. In this application of the prefix, atheism (a-theism) involves the proposition that there is not a God; and an atheist is someone who rejects (and does not merely not hold) the proposition that there is a God.