As you have discovered the InvitationID query string parameter is used to track whether or not a panelist has responded to an invitation.
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It is important to note that this does not track if a user has completed a survey. It tracks whether or not the panelist has navigated to the URL provided in the invitation email that was sent to them; this information is recorded when the first survey page is loaded.
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UPDATE: The quoted text above is out of date. In current releases of Checkbox the invitation manager marks an invitation as responded to only when a respondent completes a survey. This change was made to make the remind feature more useful.
It is not entirely clear how you are redirecting users. That being said the InvitationID query string parameter must be included in order for the invitation management portion of the application to function.