There is no recent study to back up that claim

As you (and others in the comments to that Workplace.SE answer) noticed, none of the articles cited have any citations themselves. While looking into this, I also found that it is easy to find other articles that are similar. They all mention a high percentage of people leaving within a short period after accepting a counteroffer, but none have an actual link to a source. The more honest ones don't use the hard numbers or acknowledge the lack of a source:

From https://www.wsj.com/articles/SB119431677016683480

... says Ken Stempson, director of administration and human resources for IntelePeer Inc., a communications company. He says about half of employees take counteroffers, but the majority start looking for new jobs within six months

From https://www.consultrecruitment.co.nz/blog/6-reasons-why-accepting-a-counter-offer-is-madness/

There’s an oft-quoted statistic in the recruitment world that 80% of job seekers who accept a counter offer go on to leave their job within six months anyway. Despite considerable ferreting around the internet, I can’t find the source of that fact anywhere. I suspect it might be the result of a snap poll around a recruitment office somewhere, circa 2001, and it’s become accepted lore.

I also found https://jobtipsforgeeks.com/2014/02/19/counteroffers/, which makes a very good point about secrecy:

Statistics about counteroffers are impossible to measure when you consider the interests and incentives of the parties involved. Companies that counteroffer departing employees are best served to keep that fact private, as employees may pursue offer letters just for the sake of a raise or improvement and outsiders may question if the company pays market rates.
Likewise, those who accept counteroffers may be concerned with the word getting out, as it may genuinely impact attitudes towards the employee. Employees who accept counteroffers are likely asked to keep quiet about what happened, and it’s usually in their best interest to do so.
Based on the secrecy incentives on both sides, one might assume that the prevalence of counteroffers is likely higher than reported, and success/failure rate of counteroffer is difficult to assess.

I did find two blog-post-like articles (here and here) skeptical about the numbers that say it likely came from a WSJ study from the 1960s or 1970s, but unfortunately neither of them gave a link to the study (if someone finds a link to it, I'll be happy to add it into my answer). In both cases, they came to the conclusion that everybody has just been hearing and saying the 80% and 93% numbers for long enough that it's become accepted fact.

There is no recent study to back up that claim

As you (and others in the comments to that Workplace.SE answer) noticed, none of the articles cited have any citations themselves. While looking into this, I also found that it is easy to find other articles that are similar. They all mention a high percentage of people leaving within a short period after accepting a counteroffer, but none have an actual link to a source. The more honest ones don't use the hard numbers or acknowledge the lack of a source:

From When to Take a Counteroffer From Your Employer (WSJ)

... says Ken Stempson, director of administration and human resources for IntelePeer Inc., a communications company. He says about half of employees take counteroffers, but the majority start looking for new jobs within six months

From 6 Reasons Why Accepting A Counter Offer Is Madness (Consult Recruiting)

There’s an oft-quoted statistic in the recruitment world that 80% of job seekers who accept a counter offer go on to leave their job within six months anyway. Despite considerable ferreting around the internet, I can’t find the source of that fact anywhere. I suspect it might be the result of a snap poll around a recruitment office somewhere, circa 2001, and it’s become accepted lore.

I also found Counteroffers, Secrecy, and Fear (Job Tips For Geeks), which makes a very good point about secrecy:

Statistics about counteroffers are impossible to measure when you consider the interests and incentives of the parties involved. Companies that counteroffer departing employees are best served to keep that fact private, as employees may pursue offer letters just for the sake of a raise or improvement and outsiders may question if the company pays market rates.
Likewise, those who accept counteroffers may be concerned with the word getting out, as it may genuinely impact attitudes towards the employee. Employees who accept counteroffers are likely asked to keep quiet about what happened, and it’s usually in their best interest to do so.
Based on the secrecy incentives on both sides, one might assume that the prevalence of counteroffers is likely higher than reported, and success/failure rate of counteroffer is difficult to assess.

I did find two blog-post-like articles (LinkedIn and FT Recruitment) skeptical about the numbers that say it likely came from a WSJ study from the 1960s or 1970s, but unfortunately neither of them gave a link to the study (if someone finds a link to it, I'll be happy to add it into my answer). In both cases, they came to the conclusion that everybody has just been hearing and saying the 80% and 93% numbers for long enough that it's become accepted fact.