I believe you need parameter dayfirst=True in to_datetime:
df = pd.DataFrame({'Date': {0: '1983-11-10 00:00:00', 1: '10/11/1983'}})
print (df)
                  Date
0  1983-11-10 00:00:00
1           10/11/1983
df['Date'] = pd.to_datetime(df.Date, dayfirst=True)
print (df)
        Date
0 1983-11-10
1 1983-11-10
because:
df['Date'] = pd.to_datetime(df.Date)
print (df)
        Date
0 1983-11-10
1 1983-10-11
Or you can specify both formats and then use combine_first:
d1 = pd.to_datetime(df.Date, format='%Y-%m-%d %H:%M:%S', errors='coerce')
d2 = pd.to_datetime(df.Date, format='%d/%m/%Y', errors='coerce')
df['Date'] = d1.combine_first(d2)
print (df)
        Date
0 1983-11-10
1 1983-11-10
General solution for multiple formats:
from functools import reduce 
def convert_formats_to_datetimes(col, formats):
    out = [pd.to_datetime(col, format=x, errors='coerce') for x in formats]
    return reduce(lambda l,r: pd.Series.combine_first(l,r), out)
formats = ['%Y-%m-%d %H:%M:%S', '%d/%m/%Y']
df['Date'] = df['Date'].pipe(convert_formats_to_datetimes, formats)
print (df)
        Date
0 1983-11-10
1 1983-11-10