I’ve said this in other SDMB threads about this and I’ll say it again. The best term by far is Latini.
One problem with Latine is that many people will pronounce it essentially the same as Latina. Not very helpful.
That’s why it should be Latini.
Of course this is just this non-Hispanic guy’s opinion. But it just seems like Latini is the only option other than Latino or Latina if you follow the same naming pattern. You could try Latine but different folks would pronounce it differently.
You might have mentioned the word suggested by the author of the article you linked to. It is Latinae. He chooses to employ the Latin neutral (I think it is–I don’t know Latin very well.).
“Latinae” is feminine, plural (in Latin, not Spanish). E.g. “Fabellae latinae”. But then “lego fabellas latinas”, you see how the ending naturally changes when you decline the word.
Neuter plural is “latina” in the nominative and accusative. “Latine” is an adverb, or else masculine vocative singular.
ETA Neuter is a gender, at least in classical Latin. It’s just neither masculine nor feminine, hence the name.
That sounds more like some kind of cocktail.
Thought about this the other day. We were watching something on TV and there was an ad for (IIRC) The Latin American Music Awards. Probably wouldn’t have even noticed it before, but now I was wondering, “Hmm. I wonder exactly who/what is eligible. And they didn’t use Latinx.” 
The Latin American Music Awards is a U.S.A. production, but Latin music is a nebulous category that includes various styles of music from Latin America, Spain, Portugal, and the United States as well as music sung in Spanish or Portuguese. In any case, not all of the past winners have been American at all; in fact the most awarded artist is Enrique Iglesias, who is European and has released songs in both Spanish and English.
I think I’ll start describing myself as Anglx.
The term “gringx” is already in (not widespread) use.
I’m only a quarter Latina but it’s enough to be irritated by the term. Last I read only 3% of Latinos have adopted the term, the rest presumably bothered by either the awkwardness or the imperialism that wants to correct a defining feature of their language.
Of course Latino people are individuals who vary greatly. But there is perhaps some truth to the stereotypes of machismo and feminidad. Men want to be manly, and women take pride in their gender, possibly to a greater degree than most other places. It is possible there is less demand for a gender neutral term, but I am not an expert in this area and people can call themselves what they wish.
I took salsa dancing lessons in Mexico and tried a few times to teach the basics to Canadian women, for fun, and to pass the time in bars. Several of these women wanted to lead. Now women should lead. They should play their part in leading companies, political office, families, unto salvation. But that’s not how salsa dancing works. People can do as they wish. Sometimes it is more important to enjoy.
It seems like a lot of folks here seem to think this is an attempt to be culturally respectful, and it’s failing. Whether or not it’s a good term, that’s a faulty analysis. It’s not an attempt to respect culture or ethnicity: it’s an attempt to respect gender for folks within a culture.
As a white cis dude, my opinion doesn’t really matter. The term gets used in my presence rarely, and when it’s used, it’s by a Latinx teacher I know.
White cis dudes pontificating on the subject: be aware that it’s not about you or your opinion.
“People can do as they wish” is the most important part. If two salsa dancers want to assign a lead who’s not male, why on earth should I have an opinion on the subject, or try to tell them how they should enjoy their dance?
What if two women want to dance together?
Does it matter that there are white cis-dudes who are also Latinos? Do their opinions matter at all?
People can dance however they want. But salsa dancing requires one person to lead and one person to follow. That’s just how it works, even if these roles could be (in theory) periodically rotated. If I am part of those dancing, I will assume the traditional role, partly since I am not good enough at it to do otherwise, and this is about fun and tradition, and not a sign of disrespect. It’s tough enough to learn one role passably well. (Though many Mexicans seemingly learn to dance before they learn to walk.). In Mexico, I doubt this is often questioned, but again, people can do as they please and it is not sacrosanct. I would not judge them. But the end result might not be salsa.
Fair point–I sometimes conflate race and ethnicity, and should have said, “As a white, non Hispanic/Latino/Latine/Latinx cis dude.” Better?
Cue person to come in and observe that there is “no such thing as race!” Not gonna win this, my friend! 
There’s no such thing as “race” in the same way that there’s no such thing as “money.” Anyone that wants to deny race is welcome to send me all their imaginary money as well.
Banned in Arkansas government documents; may be banned in Connecticut gov’t docs.
Sure doesn’t seem to be catching on on radio or among folk I hear. I’ve noticed Latino far more often lately, following a previous “fad” of Latinx.