No, businesses that operate in public are private businesses.
I know of no public business.
No, they are not. Read the law: “any person in any public accommodation to make a distinction, directly or indirectly, in offering or refusing to offer its services, facilities, accommodations or goods to any [individual] person because of race, religion, color, national origin, ancestry, sex, sexual orientation, gender identity, spousal affiliation or physical or mental handicap, provided that the physical or mental handicap is unrelated to [an individual’s] a person’s ability to acquire or rent and maintain particular real property or housing accommodation;”
and a public accommodation is defined by the law as “any establishment that provides or offers its services, facilities, accommodations or goods to the public, but does not include a bona fide private club or other place or establishment [which] that is by its nature and use distinctly private;”
Or as the Commision empowered by the law to enforce it says, “if you run a business that offers its goods, services or facilities to the public, you are likely to fall under the “public accommodation” provision the New Mexico Human Rights Act, which prohibits discrimination in such places.”
Virtually every business that operates in the public, offering goods or services to the public, is a public business in the sense that this law applies. “Private businesses” that sell something to the public, or provides services or goods to the public, fall under the HRA.
For an exception, you have to be able to convince the Commission and the Courts that your business is a private club. Good luck.