Hi Anthony Pica,
Allow people to continue to sign up for you webinar; change the landing page to now say watch on-demand or something similar. Now, when someone fills in the form you direct them straight to the on-demand version and give them a status of "Attended 0n-Demand". As Juli James said above, you should have that status available if your program is built using the Webinar Event program channel.
Create a new separate program to capture leads for the on-demand version. You might want to do this if the program is evergreen or will not become redundant for a while. This would be some type of gated content channel and the reason you would be building this program is to track the sign-ups to your webinar that come through your website gated-content channel rather than mixing them with the original attendees.
The answer to you question is all based around your reporting, tracking and attribution in my opinion.