Yes and no. It does come from the longform well-wishing but it has been a separate greeting for a decently long time which is reflected by the fact that "guten Morgen" became "Guten Morgen" in the spelling reform 25 years ago. (Though, they did allow both variants a decade later (as is the case with a lot of other words), "Guten Morgen" is still the recommended one.) So that means if you are wishing someone a good morning, you are wishing someone a good morning + greeting. If it is just a good morning, it really is just a good morning (for you) + greeting. I do use both variants but actually wishing someone well is more reserved for people I've been interacting with for a long time (like certain cashiers in a supermarket (if there is time) or employees of a bakery), really friendly strangers or special cases (like someone is being a dick and one is being sarcastic).