It's like this.
Foreign people are free to have their name read in whatever way they want.
Koreans today tend to prefer their name read true to the Korean pronunciation. For example, the sirname ΰ would be called "Kim" (in katakana, "Ki-mu L").
Chinese today tend to prefer (or at least think it's okay to have) their name read in the way the Japanese call "kun-yomi", which is a China-oriented Japanese style. For example, ΰ would be called "Kin" not "Kim".
In a more Japanese way of reading (we call this "on-yomi"), ΰ would be called "Kane". But this is not really a good example, because I've never heard of a sirname called "Kane".
Hope this helps.