topr8 - 2:34 pm on Mar 1, 2013 (gmt 0)

>>www.cool.example.com [wrong]

not true, many sites use www.example.example.com

and in any case there is no correct way to do it, as piatkow has mentioned it is convention that example.com resolves to the subdomain www.example.com however this is not a rule and it is not always the case.