To be clear, that article is on the Canadian QuickBooks site, and is about Canadian taxes. It doesn't tell you anything about your U.S. taxes.
The article is confusing. It says "if a non-resident of Canada renders services outside of Canada, then there are no tax obligations." But later it says "For non-residents, every payment received is subject to a withholding by the Canadian payer." I think the first statement is referring to a non-resident working outside of Canada, and the second statement is referring to a non-resident working in Canada, but it's not entirely clear. As I said before, you should ask your questions about Canadian taxes in the TurboTax Support Community for Canada. Or you might need to get advice from a tax professional who is familiar with both U.S. and Canadian tax laws, as well as the tax treaty between the two countries, which could also affect your situation.
We can't help you with Canadian tax rules here in the U.S. TurboTax community. As far as your U.S. taxes are concerned, as a U.S. citizen working in the U.S., you must pay U.S. federal and state tax on the income that you earn. It makes no difference where your client is located. Since you are a self-employed contractor, you report the income as business income on your U.S. federal and state tax returns. Since you are not an employee, you would not get a T4 or a W-2.
Whether to conduct your business as a sole proprietor or form some sort of "company" involves legal as well as tax issues. You might want to consult a lawyer who deals with small businesses. When you know how your business is structured we can give you some guidance about how to file your tax returns, but we can't advise you on how to structure the business.
International taxes are tricky and can get complicated. You might be better off getting professional advice instead of relying on what you read on the internet.
