if it also comes down to what a particular person can conceive as possibility

I think I see the problem. Whenever any philosopher says that something is logically impossible, they are specifically and openly aware that this is only conditionally true. For example, a square circle is only clearly a logical impossibility if the words square and circle mean the same thing every time I think them, including in the time period that I move from 'square' to 'circle'; if the ideas of 'square' and 'circle' actually do have a referent, and I am not just deluded into feeling a clear sense of meaning when I think those words, (and including all defining terms [such as 'right angle']); if I am correct in believing that two non-identical things are necessarily not one thing; if I am correct in believing language can refer to things other than semantic relationships; and many other fun things, including the big one: the assumption that I exist and am thinking.

However, once you have gotten everyone on the same page, and we all admit that we cannot prove that we exist, we start talking about highly conditional realities, such as those in which BattleGround God is more than a deluded memory, in which the words on the LessWrong are entered by other people, realities in which computers and circles and 'good' actually exist in some meaningful sense...

...and we don't all have the same set of conditionally accepted realities. Nor do we need to in order to make most arguments intelligible to both parties. As a matter of course, most of the things we talk about are propositioned on the existence of the external world as reported by our senses and the media. If we deviate from this, we specify this in some way.

And that explains logical impossibility... but then I realized that I hadn't read your original question carefully enough.

'It is strange to say that God is a logical impossibility, but you don’t know whether God exists. If God is a logical impossibility, then surely She can’t exist, and so you know that She doesn’t exist.'

BG has conflated "belief that God is a logical impossibility" and "you know that She doesn’t exist." The second claim should be "you believe that She doesn’t exist." BG tried too hard, and it fails. Next time, use a Real Philosopher(TM).