IMHO the real point isn't the man as part of nature part though it stems from us poking at things.
"Natural" selection and "Artificial" selection differ in that man is not (or little ) involved in the former and centrally involved in the later. But that isn't the important fundamental difference.
"Natural" selection has no end goal, no foresight, no global picture. Nothing beyond this individual member of a species and it's success or failure.
"Artificial" selection may have a very definite goal (short or long term), it incorporates the whole picture (that may be selecting individuals to increase the diversity of a rare species, e.g.). It may involve selecting for things that aren't part of the environment yet but are expected (e.g., higher heat tolerance of crops).
Yes, it requires us to do this but I don't see that as being the issue. If we were selecting individuals based on a coin flip then our effects may not be different from "artificial" selection even though we would be selecting.