Okay, he saves "the" remote page to his local computer. He modifies it to reference your script as the form action.

Then he opens the formerly remote page. That's one hit. It has not been recorded anywhere.

He submits the form. It hits your script, which updates its counter. It then "redirects" to the real page. (How this is accomplished I'm not sure, as a redirect will not post the data again. But let's assume it does, somehow, or was just a regular link.)

Then the page that's now displaying on the browser has been generated by the real web server. Now he needs to click on another link or submit another form. Since that page was not affected by your script nor in any other way updated to reference your script, it only references the original set of pages, not either your script or the original page that was downloaded. So he clicks or submits and gets a new page without interacting with your script, so no counter is updated.

From this point on, the only way to get back to your script is to enter the URL for the downloaded and manually edited page or for the script itself.

Even if you save a modified copy of each page that he was to access, only that first manually modified page will ever reference your script because all of the pages the browser renders after that first one will be the original pages that reference the original web site.