I can't really comment on the four specifically mentioned in the quote, but I can address the intent of the quote - which is basically saying "everyone, everywhere defined marriage as between a man and a woman." I think it's important to remember that today, marriage equality has been restored to the lands of the United States. Because prior to the Anglo-Americans enforcing their particular Christian view of marriage over this chunk of the continent, ideas regarding who could marry whom where considerably more diverse than just "one partner with a penis and one with a vagina."

While we have hints of non-heternormative marriages and partnerships in the east, the situation is a bit obscured in the ethnohistorical record. It's not something that European and Euro-Americans authors seemed to be privy to or particularly interested in exploring in any significant detail. So instead, we have to turn westward across the Mississippi where records on this sort of thing are a bit better.

On the Plains, marriages that our modern sensibilities would recognize as same-sex marriages were fairly common. Of course, the situation isn't as simple as that, as most Plains cultures had 3+ genders rather than just two are prevalent in American society today. There's the wíŋkte among the Lakota, the mixuga among the Ponca, the badé among the Crow, and many others. These are all third gender roles adopted by males, somewhat analogous to what we might think of as a transgender woman today. They're not exactly equivalent of course. In general, these third gender roles were liminal social positions, standing somewhere between the categories of Man and Woman - being neither, but having traits of both in addition to traits that are unique unto itself. The degree to which a third gender person could shift fluidly between Man, Woman, and Third Gender roles varied among cultures. Osh-Tisch, the most famous badé, for example generally adopted the attire of women and engaged in women's work, but when war came to the Crow, Osh-Tisch adopted men's clothing and fought with the men (earning the rather badass name Finds-Them-and-Kills-Them). Among the Crow of that time (late 1800s), Osh-Tisch's gender fluidity was considered less remarkable than the fact that a woman,The Other Magpie, also fought alongside Osh-Tisch. Like most badé, Osh-Tisch never married but did have at least one long-term relationship with a man and perhaps another with a woman (who may be The Other Magpie). When the US forced their own ideas of gender on the Crow, the people rallied in defense of the badé. Though Osh-Tisch was eventually forced by US agents to adopt what they deemed appropriate attire and labor for a man, the badé managed to keep many of the traditions associated with that gender role alive.

The situation for the wíŋkte and mixuga was a bit different. Both could and did marry men. A Lakota man might have multiple women and wíŋkte as his spouses. While there's some conceptual overlap with both men and women roles, associated with the wíŋkte were some particular social roles. They were highly regarded craftspeople, excelling at quillwork; certain rituals were under their care alone; they had unique privileges (unlike a woman, a wíŋkte could have multiple husbands). This isn't to say it was all fun and games for a wíŋkte, and they did face at least some bigotry from within the Lakota community. Hassrick's informant for his ethnography of the Lakota seems have harbored such bigotry and this fed into Hassrick's own homophobia and transphobia, tainting an otherwise informative book.

And lest you think, third genders were only for males, there are plenty of examples females expressing alternate gender identities as well. Unfortunately, these get less press than male third gender roles. An interesting example is Masahai, a Mohave hwame:, who was married three different women early in her life, two concurrently. Later in life, after having been separated from the wives, Masahai was married two men.

EDIT: I had a request for a source, but the comment asking for that appears to have been deleted by its poster (my mod-vision isn't detecting it anymore). So for the benefit of anyone who's interested I thought I'd just add my replay to that post here.

I was mainly working with Changing Ones: Third and Fourth Genders in Native North America and The Sioux: Life and Customs of a Warrior Society. Two-spirit People: Native American Gender Identity, Sexuality, and Spirituality is also a great source on the topic, but I don't have a copy handy so I didn't incorporate material from there.

I just found an article by an expert discussing Aztec marriage. Multiple wives were the norm among the wealthy.

I'd also like to add that calling all of these diverse practices "marriage" is a translation problem. When we see things that involve a long-term pair bond, we translate the name of it as "marriage". So we're looking for things that look like European traditional marriage, and then referring to them as marriage, even though in a given culture, it's not an exact translation. Or there's not an exact correspondence.