According to the book Newspapers of Record in a Digital Age: From Hot Type to Hot Link by Shannon E. Martin and Kathleen A. Hansen, there are two ways that the term "paper of record" is understood. The first is in a legal sense. Most states, counties, and municipalities have laws that require certain types of legal proceedings to be published in a local newspaper as a legal notice. These often include such notices as probate cases, divorce proceedings, certain cases of debt collection and liquidations, class action lawsuits, and so on. The requirements of what proceedings must be published are pretty variable. For instance, New York state is one of three states that requires newly-formed Limited Liability Companies (LLCs, PLLCs, LPs and LLPs) to publish an announcement of their formation in two local newspapers once a week for six weeks in a row.

The municipality, or county, or state will have some rules in place under what kind of newspaper qualifies as a "paper of record", usually having to do with circulation and frequency of publication. In New York state, these are referred to as "designated newspapers". The newspapers themselves actively try to get the designation, which they can get by filing paperwork with the county annually. Because of the legal requirement to have these legal notices printed in the paper, the litigant (be it the government, corporation, or private individual) is responsible to pay the cost to the newspaper of running the legal notice. There are usually rules in place for how much the newspaper can charge, and while they are cheaper than a typical commercial advertisement, they still amount to free money for the newspaper.

You can see one of the laws in New York governing these legal requirements here. And here you can see a list of the current "designated newspapers" in Westchester County, just north of New York City, that can be used to fulfill the requirements for legal notices.

In New York City itself, according to the Clerk of Courts website, you actually have to call or come to the office to get a list of the designated newspapers. But no matter, because, according to Queens County's website:

"The Queens County Clerk’s Office has a rotating list for the assignments of newspapers for publication. All publication directives are assigned in the order that each request for publication is received. You may request your publication designation in person or by mail."

Presumably, because the list of "designated newspapers" is so long in New York City, the city dictates to the litigant which newspaper(s) they have to publish in, so that the papers aren't trying to undercut each other on price and so that they each get a piece of the pie.

The New York Times, in their local edition, is one of the newspapers that runs these kinds of legal notices. Under the legal standard, then, the New York Times is a "paper of record" and has been a "paper of record" longer than nearly all newspapers still operating in New York. (The one exception being the New York Post, though I am not sure if they are a "designated newspaper" or not.)

The second standard for a "paper of record" is the more colloquial one. It's defined more by its reputation among historians and librarians, according to Martin and Hansen. The New York Times has made a point since its inception of documenting all the most important local events in the city of New York, big or small. Of course, it's not possible to do so completely, but in addition to their national coverage and political coverage, they have tried to also operate as a nonpartisan local paper for New York, both when New York City was only Manhattan and now that it contains all five boroughs. Though it may be somewhat arguable, they have done a better job of that than any of the other New York newspapers, who (especially in the 19th century) were more concerned with filling their pages with headline-grabbing and/or national and international material.

According to Martin and Hansen, the usefulness of the New York Times and what led to it initially gaining its reputation as a "newspaper of record" the annual index of all their articles they published over the preceding year, beginning in 1913. This made its archives easily searchable by subject. This index was distributed to libraries throughout the country, and the index has been maintained to the present day. I'll just quote Martin and Hansen directly, to give the details of how the reputation and term came about:

The [New York] Times was the first newspaper to publish continuously, beginning in 1913, an index of subjects found in its pages, and as a result, librarians began using the expression “newspaper of record” in reference to the Times soon after the paper’s index was available. (The added value of archival paper and the indexing service, along with the editors’ intentions of community record keeping at the New York Times, was used as part of a promotional effort for library subscriptions in the 1920s.) By 1927 the Times used the expression, too, in promotions for its rag paper edition published for the purposes of library archiving. That same year the Times organized a contest around the slogan “The Value of the New York Times Index and Files as a Newspaper of Record.”

Though the New York Times is a traditional reference resource for libraries, it is not the only daily newspaper considered by many librarians to be a “must have” for reference collections. Other newspapers indexed and widely collected by libraries include the Washington Post, the Los Angeles Times, USA Today, the Wall Street Journal and the Christian Science Monitor.

So, if the New York Times did not directly invent the term, they certainly capitalized on it early on, once their newspaper was being referred to that way by librarians, historians, and archivists. This sloganeering has perpetuated their reputation as being one of the preeminent "newspapers of record" published in the United States.

Generally speaking, then, a "newspaper of record" doesn't preclude a newspaper from employing an editorial staff or even editorializing their coverage. It just means that the newspaper has a national reputation, includes broad coverage, and is useful in the pursuit of historical knowledge by librarians, historians, archivists, and others who may need to reference events of the past. By those standards, the New York Times has earned its reputation by most assessments, but that doesn't mean it's singularly valuable, either. It does have competition. As Edward D. Starkey wrote in 1988:

"Although The New York Times is not a national newspaper for the United States in the sense that Le Monde is for France, even its critics concede that it is one of the keystones of North American journalism, and in fact most public and academic libraries have a current subscription, a back run on microfilm and the index. The danger lies in taking too seriously the Times' claim to be the newspaper of record for the country and overlooking other newspapers of national stature and regional importantce."

Starkey cites the Christian Science Monitor, the USA Today, the New York Amsterdam News, the Wall Street Journal, the Toronto Globe and Mail, the Philadelphia Inquirer, the Philadelphia Daily News, and several computer index services (some that still exist like NewsBank, but several that do not) that were useful as of 1986 for archivists and for history-gathering purposes.

Sources:

Martin, Shannon E., and Kathleen A. Hansen. Newspapers of Record in a Digital Age: From Hot Type to Hot Link, 1998, pp.1-14.

Starkey, Edward D. "A Selective Overview of Newspaper Indexes - 1986", published in Newspapers in the Library: New Approaches to Management and Reference Work, ed. by Lois N. Upham and Peter Gellatly, 1988, pp.133-142.

Further reading:

Black, John L. C., and Michael A. Wineburg. "Publication of Legal Notices in New York: Guidelines for a Revision", Cornell Law Review, 1969.

Mills, T.F. "Preserving Yesterday's News for Today's Historian: A Brief History of Newspaper Preservation, Bibliography, and Indexing", The Journal of Library History, 1981.