FOX News and right-wing media are already branding Democratic Socialists as "socialists"...

[still_one]

5. She is a member of the Democratic Socialists of America (DSA).

"Jeff Stein wrote about the DSA in 2017 so Vox could clear up some misconceptions.

What does DSA believe in?
Like most socialist organizations, DSA believes in the abolition of capitalism in favor of an economy run either by “the workers” or the state — though the exact specifics of “abolishing capitalism” are fiercely debated by socialists.
Stein gave more specifics:
In practice, that means DSA [members advocate] ending private ownership of a wide range of industries whose products are viewed as “necessities,” which they say should not be left to those seeking to turn a profit. According to the DSA’s current mission statement, the government should ensure that all citizens receive adequate food, housing, health care, child care, and education. DSA also believes that the government should “democratize” private businesses — i.e., force owners to give workers control of them — to the greatest extent possible.
But DSA members also say that overthrowing capitalism must include the eradication of “hierarchical systems” that lie beyond the market as well. As a result, DSA supports the missions of Black Lives Matter, gay and lesbian rights, and environmentalism as integral parts of this broader “anti-capitalist” program.
“Socialism is about democratizing the family to get rid of patriarchal relations; democratizing the political sphere to get genuine participatory democracy; democratizing the schools by challenging the hierarchical relationship between the teachers of the school and the students of the school,” said Jared Abbott, a member of the DSA’s national steering committee. “Socialism is the democratization of all areas of life, including but not limited to the economy.”

https://www.vox.com/policy-and-politics/2018/6/27/17509604/alexandria-ocasio-cortez-democratic-socialist-of-america

That is how she identifies herself.

How she wants people to characterize what she means by that is going to be up to her, but regardless, because she has chosen to be part of the DSA, and appears to embrace the ideals of that organization, she has introduced it into the conversation, whether the connotation is positive or negative to people it is up to her to explain what she means by it.

Because she has identified herself as a Democratic Socialist, what that means will come in the conversation, and it won't just be the faux or right wing outlets.