Saying you're a programmer is like saying you're a writer. It's not an overgeneralization. It's a broad field. There are technical writers, business writers, novelists, journalists, poets and hacks. Some write soaring prose, others mundane workaday text. They write in different languages. They right for different audiences and cultures. There are amateur writers and professional writers. Those who write for love. Or the love of words. Those who write for money. They're all writers.
To me, programming is a lot like writing of other forms. And in more ways than this. So it is an apt comparison.
PS this reminds me of advice I was given leaving school.
"So what makes a true writer?"
"Well, writers write. Plenty of people say they're writers, or want to be writers, or criticize other people's writing and say they could do better. But the thing that sets a true writer apart is that they actually write, rather than talk about writing."
Plenty of people talk about being programmers, or say they know how to program, or critique others' programming and efforts. But the mark of a true programmer is easy to find.