Smalltalk and professional exchange aren't politics, the author is conflating the too.
When people say "no politics" they mean have a position on certain issues that mostly are off topic in a business environment.
For example if I have my group where nobody is religeous. You could rant about how stupid religeous people are because nobody would feel particularly attacked and some would nod along. Disregarding that the pittyful self-revelation from pointing at others calling them stupid, this is a political stance.
But we employ people from all over the world and viewpoints change. Some don't have the most dense main stream belief you find everywhere. You don't go into the next office and pronounce how atheism is the best thing. That is meant with "no politics". It is a requirement for multi-cultural exchange without immediate conflict. It is of course not restricted to religion.
The auther misunderstood what politics means. What he describes is office and relationship dynamics. There is quite a bit of overlap, especially when it comes to signal your viewpoints and perspectives in the hope to get recognition. I would be careful about that in a professional environment though. Depends on the company and how many cultures meet each other in random watercooler talk.
You can convolute the terms here, but it just blurrs the precision of any statement.
That said, relationship dynamics or "power play" leads to an effect where the most competent people often aren't the most well liked people. That is unfortunate and not very new. But the problem cannot be adressed by "talking more about politics". On the contrary, it would make things much, much worse.
That's politics in the office, not office politics.
But you are right that the author is mixing things up: Office politics isn't collaboration as described in the article. Office politics refers to things like one-upmanship, taking credit for stuff, playing the blame game - making yourself look good and others look bad, to get raises or promotions. Or for a phrase used in the article, office politics is about becoming a scheming backstabber.
It doesn't have to be.
It can also be the opposite.
Making yourself indispensable. Being the one who shows up for people (not as in "comes in and does a lot of unpaid work", but as in "helps out when other people need it"). Giving people credit where credit is due, especially the unsung heroes.
If you are well-known around the office as the person who is honest, kind, and helpful, the next time someone else tries to take credit for your work, make you look bad, or otherwise stab you in the back, it's much less likely to work—and when that kind of thing fails, it invariably makes the person who tries it look much, much worse.
That's not politics, that's just being a good person/employee. Politics is when you try to look that way, no matter the cost.
Often times, doing the good stuff and building credit for that takes a long time and the right environment to identify and credit it.
Playing the politics game is much faster/easier and leads to quick results, because all you have to do is be visible as much as possible and manipulate a little bit here and there.
People aren't great at identifying the career manipulator and in the short term will give those guys the promotion/responsibility.
Which you have to do if you want to get ahead, no matter the quality of your work. Merit and competence are only weakly correlated with salary.