Not really; barratry is more a DoS on the system of justice (it wastes court time), however certainly some barrators could focus exclusively on one private party. Making false complaints to the police is not barratry. Maryland, which is the one of the States explicitly mentioned in the article, has a specific and atypical statutory definition[1].

It is hard to be a barrator in modern systems of justice based on English common law. Defendants of repeated claims by a single claimant can get relief from the courts, even if the claims aren't strictly frivolous.

In common law a barrator is someone who repeatedly brings frivolous actions to court, knowing they are frivolous. There are statutory definitions of barratry as a criminal offence, mainly in several of the States of the U.S.A. The usual requirements for criminal liability apply, and there is usually a threshold value for the number of unlawfully-brought actions, so there is ample opportunity for courts to impose a graduated response.

In most common law jurisdictions, it is essentially impossible to bring a frivolous lawsuit as licensed persons (lawyers of various types, paralegals) will generally not risk professional liability, and court clerks generally have the power to reject proceedings initiated by unlicensed persons if they are manifestly frivolous. In other jurisdictions, lists of ineligible litigants are maintained[2] and updated from time to time by the overall system of justice.

Additionally, most courts usually exercise their power to make a costs order against someone who still somehow manages to progress frivolous litigation, usually on an indemnity basis, which can be sufficiently expensive when the other party or parties has had to hire legal representation that it is strongly dissuasive. Most potential barrators could not afford to reach the threshold of criminal liability, because liability for the costs of the court and defence (and collecting those costs) exceeds their ability to pay.

However, access to relief from the courts is important, so there is generally a mechanism to allow such people to bring cases which have some significant probability of succeeding to court nevertheless.

Unfortunately, civil lawsuits are usually settled out of court, so courts may not see abusive patterns (courts are generally shy to inquire into consensual agreements between adversarial parties) until a defendant refuses favourable on-the-steps-of-the-court settlement offers instead opting to reach a final judgment and a trial on the matter of costs. It's then that a vexatious litigant that casts its net widely (i.e., filing many dubious different courts because it can, thanks to corporate scale or whatever) is most likely to be "discovered" by a court for the first time.

Harassing behaviour by creditors or their collections agents is a different matter, and threatening to bring obviously frivolous proceedings does not typically incur criminal liability -- it's only the act of bringing such proceedings, knowingly and repeatedly, that will reach the threshold of barratry (where that still exists as a specific offence) or its rough equivalents. Most "private prosecutions" are those brought by a public body with the explicit statutory power to do so, where the body is something other than the usual prosecutor.

Finally, it is exceptionally rare that barratry or vexatious litigation could escape the domain of private law. Criminal prosecutions are generally brought by a public authority, and it is exceptionally rare (even in systems where it is even possible) for a private person to bring a criminal prosecution. It usually requires permission from the court or a public authority. (Even in the pre-modern system -- here roughly when professional public prosecutors were granted absolute discretion over bringing any criminal prosecution -- which depending on jurisdiction ranges from the 17th to 20th century, a private prosecutor would still have to seek an indictment, and malicious, frivolous, or vexatious prosecutions would almost inevitably fail to do so). Threatening to report innocent behaviour to police, while harassing and not in the public interest, is not the same as actually engaging in malicious prosecution.

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[1] https://law.justia.com/codes/maryland/2010/business-occupati...

[2] https://www.gov.uk/guidance/vexatious-litigants -- all of these judgments are generally easy enough to find online; an example is https://www.infotextmanuscripts.org/vexatiouslitigant/vex_li... -- not all of them would have been barrators under the common law (before statute superseded barratry and many other common-law offences in England and Wales).