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Using a list for this purpose is not ergonomic, which is very noticeable in the toString implementations of your two approaches (the second one is spot-on). For a reasonably fast solution, a char[] is clearly the way to go.

No need to name it characterBuffer, you don't normally put the type's name in the name. So buffer is nice and simple for this purpose. A StringBuilder implementation will probably not need any other kind of buffer anyway.

As @maaartinus pointed out, the biggest problem in both of your implementations is the throw-away char[] allocations. Be careful with that, it's a big mistake.

Perhaps the approach they were looking for is the technique taken by the JDK's StringBuilder implementation: using String.getChars to copy characters from a source string to a target char[] at some offset and with some length, like this:

 if (str == null) str = "null";
 int len = str.length();
 ensureCapacityInternal(count + len);
 str.getChars(0, len, value, count);
 count += len;
 return this;

Now that looks unbeatable in terms of performance.