Oh, oo, and oh (a question)

[Roses Ablaze]

Is there a difference between those representations of Japanese phrases (ie do they indicate different things or represent different sounds and/or characters) or is it just a preference? Would it be appropriate to spell Kouji Kohji, etc? (the reason I'm asking is because the 'oo' always bothers me because it represents the 'u' sound in English and I was wondering if it's appropriate in all cases to replace it with one of the other two).

 

[Zhen Lin]

< oh > is so-called passport romanisation.

< ou > is wāpuro romanisation, or as I tend to find it, otaku romanisation.

< oo > is JSL romanisation, and also used by linguists.

The correct way to do it in Hepburn-shiki and Nihon-siki is < ō >. The correct way to do it in Kunrei-siki is, apparently, < ô >.