Alex, this is fairly misleading and I think you know it. There is nothing to prove on AirBnB!
The mobile listing did not mention a twin bed + cot. I read the listing and even gave it to my fiance to see if she liked it as well -- no mention then. I double-checked again when I left the review -- no mention then. I gave you feedback that you should update the mobile version of the listing to include this information, and you have done that. I am glad to see that it won't be a problem in the future.
I made no complaint about paying a price difference, though it would have been a nice gesture on your part to cover the difference. The error leading to the misunderstanding was on your part (or perhaps the AirBnB mobile team's part), so your taking responsibility for that would have been the "cooperative spirit" that you mentioned.
Finally, I'll leave you with another piece of feedback, since you were also misinformed about your ability to cancel a reservation. You told me that only I could cancel, but since the reservation was made within 24 hours, that it would not be possible to cancel. You did not offer to cancel, and you did not offer a refund. Both are possible, and you would know this with even a basic search of the site: https://www.airbnb.com/help/question/166
So in the "cooperative spirit" you mentioned, it would have been nice to (a) acknowledge that your listing was not described properly, (b) offer to cancel in light of that fact, and/or (c) offer to cover the $10 difference in price in the event that you couldn't afford the cancellation. You decided to deny any omission on your part, claim to not be able to cancel a reservation, and refuse to honor the difference in rooms. This is not the cooperative spirit of the airb&b network, in my opinion.