How can I correct the correlation names on this sql join?

Use table aliases for each reference to PoliticalFigures instead:

SELECT 
  Countries.Name AS Country, 
  P.Name AS President, 
  VP.Name AS VicePresident
FROM
  Countries
  LEFT OUTER JOIN PoliticalFigures AS P ON Countries.President_Id = P.Id
  LEFT OUTER JOIN PoliticalFigures AS VP ON Countries.VicePresident_Id = VP.Id