That was my first thought too, but I don't think that is the problem. When I intersect, I tell the program to keep clip-tags that intersect the window, so my output is 7,267 clip-tags. I know it is the clip-tags because that file has other columns that the window file does not and the output matches the format of the clip-tags bed file.
When I use count, the output is the window coordinates, with a new column: the number of clip-tags that intersected the window.
That is why I'm confused. If count tells me that 12,866 tags overlap this window, but then when I intersect the clip-tags on the window and tell the computer to get rid of any clip-tags that are not in the window, it only gives me 7,267 clip tags. So why were the rest filtered out?