I never said you have to have a fixed peak for SJ - you don't. What I meant was if you watch top level SJ you will rarely see riders not wearing a fixed peak. However if you watch eventers most will have a skull hat with a silk for SJ. At lower levels you see a mixture - but always hats without a fixed peak for XC.
It is up to the show organisation (ie BS, BE, RC, PC etc) to stipulate what is required and that will normally be the PAS015 or EN1384 etc standard. Then there may be an extra stipulation saying fixed peaks are not allowed - which is always the case for XC. You should always check the rules. For SJ (unless stated otherwise in the rules) either type is 'correct'.
So long as you are following the rules it's personal preference which type of hat you have. Plenty people have both types. Same goes for body protectors, they are only mandatory for XC but many people wear them for SJ where they are optional.