The phrase "profession carried on in a state" typically refers to a profession or occupation that is conducted or practiced within a specific state or jurisdiction.
North Carolina does have a convenience of the employer rule as well. See What is the convenience of the employer rule?
I wonder if you were traveling to NC and decided to relocate there to make the job easier. If you go into work at the office every so often, you are part of the NC work and income. Working under NC conditions and rules makes MC income.
If you never came into NC and telecommuted for the employer's convenience - they didn't have an office in your area, etc you would not be liable.
An engineer example, if you have a PE in Texas, your income would not be taxed in NC simply because the office sending you work is there. If you have a PE in NC and happen to work in TX, then you would have NC income from your state profession license.
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