Er, no. You don't have
$visit = $SESSION['visit'];
on the first page (where $visit is getting its value from $HTTP_GET_VARS (or $GET)), because it's not in the session yet. On that page it would be more like
session_start();
...
...
$visit = $HTTP_GET_VARS['visit'];
...
$_SESSION['visit'] = $visit;
And the ... is of course just short for "some code" which I didn't write because it's irrelevant to the point.
And as I said, on every other page where you want to use this variable, you'd have something like
session_start();
$visit = $_SESSION['visit'];
$_SESSION is much like any other array, you can add things to it, you can read from it, you can delete things from it. It just has this magical extra ability to keep its values from one page to the next.
Like I said, read the manual; that will fill in the bits where I'm handwaving.