The bwst way to write down my English name

I would go with the first - list your name using the correct spelling with the nickname in parentheses.

I prefer; Min (Minhyuk) Lim. If I saw this, I would probably call you "Min" if our relationship was casual business. If we were working together closely I would likely ask you what you prefer. But, Min is a fairly straightforward name that is easy for us to pronounce.

The first three options are a bit confusing... I would question how you wanted to be called. I would probably ask you. These options would be more likely to lead to someone asking about your name. I don't know if this is a good thing.

The last two options seem like are you changing your name (at least professionally). There may be benefits to this strategy.

I prefer the fourth option. It says to me "you can call me Min, but my real name is Minhyuk."