Well, TN is none of those, and here I am authorized to work legally. If I apply for another job elsewhere, the honest answer to this question is yes, despite the fact that I will be required to get a new TN.
> The nonimmigrant USMCA Professional (TN) visa allows eligible citizens of Canada and Mexico to work in the United States as USMCA professionals in prearranged professional level business activities for U.S. or foreign employers.
Colloquially people refer to a status granted through a particular visa category as "visa" e.g. "I am studying on F-1 visa", "H-1B visa employees" etc.