What I think the Swedish mean is that 'artist' is a general word for any artist (actor, singer, musical star, painter...) and that 'konstnär' is used more for the specific arts of painting, sculpture, installations... The same difference is used in Dutch, German and maybe some other languages.
specifically when it comes to professions, you generally don't use a/an. while in English you would say "I am a lawyer" you would say "jag är advokat" in Swedish to mean the same thing. If you use "en" in this case, it sounds like you're saying "my career as ONE artist" rather than "an artist".
But the English translation doesn't work. You would only use 'My career as an artist' if you were following it with something else. It is an opening of a sentence only and makes no sense otherwise. I tried 'My career is as an artist' but no, that was marked incorrect. That means, that no, it does not mean 'I work as an artist' in English. It means 'as an artist....I do/teach/sell/beieve/something else' there is no inference of working as one specifically. So, who knows?