Hi Jhargrave, the reminders are specific to a given email distribution, so any reminder email you send based on the second email distribution will only take into account contacts who have not completed the survey based on the second email distribution. In other words, any contact who completed the survey via the original email distribution would not be excluded from receiving the reminder for the second email distribution.
If you configured your survey to prevent multiple submissions, then any contact who has already completed the survey (i.e. from your original email distribution) and clicks on the survey link in the reminder for your second email distribution will simply see the page indicating their response has already been captured.
One simple option to mitigate confusion for your contacts would be to simply adjust the copy of your reminder email to make it clear that anyone who has already responded to the survey need not do so again. Sometimes being honest about a simple operational error is the easiest way to handle the situation.
Hope that helps.
Cheers,
Cameron

Another option is to remove the contacts who already submitted responses to first invitation from the second invitation's contact list, so when the reminder sends, they will not be included.