Memilih rentang tanggal MySQL dengan date_format

Anda harus menggunakan STR_TO_DATE karena Anda ingin mengonversi string kembali ke 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