Domain privacy has zero effect on SEO
This has been confirmed by Google
here
Actually it can have an affect on SEO and the trust Google has in a site. The article that you posted was more than 2 years ago (not that is was accurate then) and in the SEO world that is a lifetime. Heck a year can be a lifetime.
Private registration can be used as a factor in certain situations and especially when used in conjunction with other factors as has been stated by Matt Cutts of Google and has been stated from other Google sources as well. Look for recent (a year or less) information on this and you will see that it can be used for ranking in some cases.
They look at private registration in some cases as people trying to hide something. Google likes transparency and openness and they can see private reg. as hiding something. They base SEO rankings in-part based on their trust.
Some reference:
Matt Cutt's suggested that private Whois data can be combined with other factors as a negative signal.
https://www.mattcutts.com/blog/2006-pubcon-in-vegas-thursday-site-reviews/
http://backlinko.com/google-trustrank
Public vs. Private WhoIs: Private WhoIs information may be a sign of something to hideâ€. Matt Cutts is quoted as stating at Pubcon 2006:
https://www.linkedin.com/pulse/googles-205-ranking-factors-infographic-chauhan-50k-connections
https://www.hermish.com/5-weird-negative-ranking-factors-and-what-you-can-do-about-it/
These days Google being the giant entity it is has access to whatever information they want and can easily find out the data behind the domain privacy as stated in this video.
Google doesn't have free access to private registration data always.
Here is what is found on ICANN about private registration and releasing that information to other registrars who are not the registrar of a domain:
"These services are independent of ICANN but regulated by them through policy. ICANN does require that registrars send private registration data to the shared registry,
meaning that ICANN has access to that data, but they only reveal it to other registrars when necessary for verification, such as in the case of domain transfer. Note that many registrars will also reveal private registration data on a case-by-case basis upon receipt of a cease-and-desist letter, DMCA takedown notice, or other exposure to legal action. ICANN leaves these decisions to the registrar."
There is tons of SEO information out there and one person says one thing and another says the opposite. It's hard to know what is or is not accurate because the SE's aren't going to give out all that information. There's even a quote somewhere out there from a Google employee (not sure which) that says that the number of people that know all of Google's ranking factors is less than the number of fingers on one hand. They have certain people know one thing and others know other things.
SEO is based a lot on speculation. Since Google is on record as saying that private reg. can be used to decide trust which affects SEO rankings then having non-private reg. will have one less thing that can possibly be used against a site negatively.
Do I believe that Google can't see private whois data like ICANN says? To be honest, I think that they can get whatever data they want. However, their algorithms won't have open access to that data I know. ICANN won't leave the door wide open. BUT when there is someone looking at a site for a possible manual action against a site, I am pretty sure they can find the information they need.
There are many influential and know people in rankings including Brian Dean from Backlinko and Neil Patel just to name a couple do not use private reg as they are believers of what Matt Cutts and other Google employees have said can be used as a factor.
I like to use private reg. most of the time for spam protection and just because I don't think everyone needs to know that information most of the time. If I strongly believed private reg. was a big SEO factor then I wouldn't use it unless I absolutely had to.
Each person has to judge for the selves if they want to risk the importance (if any) private reg. affects SEO and what value they hold for it. It's all just s judgement call.