The entertainment industry. The music business. TV, theater, movies, recordings, the internet, public performances. All of them are venues for performers.
Musicians may or may not like the idea that they are in the entertainment industry. They would probably acknowledge they are in the music business. They probably don’t call themselves performers, but, of course, they are. The names that people give themselves are quite varied.
Actors includes actresses (a woman can be called an actor) but not the reverse. Singers might be actors as well as vocalists, but not always. Vocalists should be actors but they don’t always understand that. It also depends on what you think “acting” is. In theater, it isn’t about acting at all. Confused? It’s OK. You have to be in theater to understand.
People who are “celebrities” (meaning people who are famous — sometimes — as with the Kardashians, for being famous) don’t have to be anything other than famous. Silly? Yep. But very true. These days you can become “famous” by having lots of followers on YouTube.
Where, in all this, is there room for those who are shy? Reluctant? Retiring? People who are introverts? Should they be in any aspect of this world? No. But they are often there, surprisingly, and sometimes do quite well. How can you be in “show business” (or, the business that makes shows) or the entertainment industry, or be a musician, and/or a vocalist, or any other of the above listed jobs? and be uncomfortable? I do not know. I am not in any way introverted nor shy and I have never been a quiet, stay-at-home iconoclast. I have known others who were not extroverted who can be very uncomfortable in front of an audience but quite comfortable doing whatever they do.
The advice I would give someone who wants to be involved in the “entertainment industry” is simple: don’t get involved in any part of it that involves performing in a live venue in front of an audience. Work in a studio, or behind the scenes, or in some other aspect of the work like composing, writing, producing, or promoting the work you like or do. If you want to be a successful singer, find some “holy boldness.” ad go for it.