enigma1 - 9:57 am on Nov 22, 2011 (gmt 0)
If they search for the exact domain name they may not be interested on the site's content. You would think if someone looks for examples he would type-in example not example.com. Or he may go directly to example.com if he's been there before.
For the other page is this something you see on various pages with many visits. Exact keywords match and they leave in few seconds? Any problems with the html and different browsers? Is the page content unique or it contains some text too common the visitor already knows about it.
I also don't know if the statcounter tracks re-entries. Sometimes I would read a page, then search again and re-enter the same page and then I will leave right away as I already read the description. It's not something that happens often though.
Can they even track how long visitors spend on a single page?
Assuming js and cookies are accepted by the browser, then yes. The first request won't be able to track anything, because it's the earliest the cookie can be sent from the server but with ajax you could make various requests from the client to the analytics server once the page is loaded and therefore tracking cookies could do it.
Now on the server end if the code never sees the ajax request, it will not record anything and may imply the page was never loaded successfully. The few lines of js analytics code you integrate on an html page can load other code to do the job but of course each js analytics program maybe different and not coded in the way one would expect. For instance if they record the very first click but don't verify it with the exit you won't know if the page was properly loaded.