Is there a significant difference between "Jihun" and "Jihoon"? If the Korean ㅜ sound is supposed to sound like "oo" in English, spelling the name here "Jihun" makes the vowel sound more like the "hon" in "honey". I've never seen 지훈 spelled like that.
I think that spellings for names should include several of the common romanizations. Name-spelling from Korean to English is not the most important concept, so as long as it generally corresponds with the Korean sounds, it should be accepted.
No. It is completely arbitrary and based on how an individual chooses to spell their name. The major transliteration systems have guidelines to use for historical figures or people who don't who haven't designated a preference but they still all make allowances for what has been accepted as the norm, such as Syngman Rhee. Koreans will insist they are the sole arbiters both of how Korean names must be spelled in English and how English names must be spelled in Korean but there is little consistency. For instance, 이 is romanized as both Lee and Rhee (along with a couple of other variations) based on familial preference. For the answer to not accept both Jihun and Jihoon as well as Jihuna, Jihun-a, Jihoona, Jihoon-a, Ji Hun, Ji Hun a, Ji Hoon, etc.
We follow 로마자 표기법, or the Romanization Rules, by the NIKL. Maybe 지훈 happened to be a simple name, but for example 재섭, my name, could be spelt in tens of different ways. You can always see the romanization we use on the drop-down hint, so there doesn't have to be any confusion.