Your post says there are two options when a form is complete:1.

Your post says there are two options when a form is complete:

1. Show a Thank You page

2. Redirect to an external link

I really wish you could add a 3rd option

3. Reload the same form in the same place.

That is possible, simply paste the direct link of your form, or the URL of the webpage where your form is embedded, inside the external link option(2):

You can even show a Thank You page, and then autoredirect to your form:

https://www.jotform.com/help/89-Create-a-Custom-Thank-you-page-with-Auto-redirect 

Let us know if you need more help.