And how does an "emerging composer" turn into a straight up "composer"? By finishing a doctoral degree? With a tenure track teaching position or other substantial music-related gig? Upon receiving a commission from a major orchestra or signing with a major publisher? A big award such as a Guggenheim fellowship or Rome Prize? I have no idea!I keep forgetting my passwords to the various sites I read, so I'm posting my answer here.
The whole "emerging composer" thing is a construct. When my music first got published, I sort of thought of myself as something like an "emerging composer" (but that before the snappy term--that really means nothing--started being used). Now that I have achieved the level of recognition that I will probably have for most of my life (miracles could happen, I suppose), I am no longer an emerging composer. I'm just a composer. I know many older composers (older than me) who have had music performed and recorded by lots and lots of people and published by lots and lots of publishers, who have never "emerged" because their names have never become household words. They are just composers too.
The handful of composers who become household names in the classical music world are people who have excellent people doing excellent publicity work for them. Some of the people just under "household" status do an excellent job of doing publicity work for themselves. Some are affiliated with institutions, and some are not. There are composers who are able to raise money to pay publicity people (and there are composers who must have access to personal fortunes, but they never talk about it), and there are composers who have spouses or partners who do the necessary publicity work gratis, but money (in some shape or form) is still a factor in their "visibility."