In every article or book mentioning him that I've come across, it's always written in the Western order, "Syngman Rhee" rather than the Korean order, "Rhee Syngman". His contemporaries like Mao Zedong and Kim Il-sung would be referred to as exactly that. Likewise, I've never come across "Chung Hee Park".
If you look at Korean sources, his name is spelled and spoken in that convention, Rhee Syngman. As for why it is spelled Syngman Rhee in English, there are a couple reasons. One is that his name is western. In East Asia during the past, people with a French, English, etc first name would sometimes invert their name order to conform to western style. Second, it is common for Asians living in the west to invert their name order to conform with the custom there. Rhee lived in the US for several decades and was well known to policymakers by the time he took power as Syngman Rhee. Third, since these names are written in characters, each person has a bit of leeway in how they prefer their name to appear in English. You mention Park Chung Hee above. But when he first visited the US in 1961, Americans officials were spelling it 'Pak Chung-hui' in documents, using the McCune–Reischauer conversion. It wasn't until he told them that he preferred Park Chung Hee that it becomes standardized in US documents and then the press, etc.