Debe usar STR_TO_DATE
ya que desea convertir string
volver a date
SELECT MvtDate, date_format(MvtDate,'%d-%m-%Y')
FROM `immmvt`
WHERE MvtDate BETWEEN STR_TO_DATE('01-01-2010','%d-%m-%Y') AND
STR_TO_DATE('02-01-2010','%d-%m-%Y')
FYI: DATE_FORMAT() converts date to formatted string representation. STR_TO_DATE() converts formatted string back to date