We use “名人” for the master.
When we use “名人”, we don’t care if the master is “有名人”.

The meaning of “名” of “名人” is guessed that it was equal to “名” of “有名人” in earlier.
As you say, “名” of “有名人” is famous.
It implies “well-known-in-public”, and “good”.
Especially, the former is quite important for “名” idea.

But now “名” of “名人” means just “good”.
It doesn’t need any public reputation.
I think this meaning is derived from the original one.

In this meaning, there are some other words.
“名曲” “名言” “名器”…
These words don’t need public reputation, especially the third.

I can add one more example about “mei”.
“名人(meijin)” means “master”.

“彼は　チェスの　名人　です” means “He is a master of chess”.
We can use “名人” although he(she) is not “有名人”.
I guess in earlier meaning “名” of “名人” was near to “名” of “有名人”, the meaning has changed.