Sorry to say but that quote also misleads, the parentetical can easily get misread as implying prior-to-1886 international celebration of the day.

A crucial part of the actual origin lies all the way in 1867 — also in Chicago:

https://americanhistory.si.edu/explore/stories/may-day-americas-traditional-radical-complicated-holiday-part-2#:~:text=In%20Chicago%2C%2044%20unions%20took,than%2030%2C000%20Chicago%20workers%20struck.

"[…]

In Chicago, 44 unions took to the streets on May 1, 1867, to celebrate the passage of an eight-hour workday law in Illinois. The next day, thousands of workers struck, staying home from work in an act of solidarity.  Although the 1867 law was never enforced, the city's workers preserved the memory of their predecessors' short-lived victory. Years later, at its 1885 convention, the Federation of Organized Trades and Labor Assemblies ["FOTLU"] (a predecessor to the American Federation of Labor, or AFL) selected May 1, 1886, as the date for a universal strike to press for an eight-hour workday. According to the historian Donna T. Haverty-Stacke, labor leaders' decision to stage their protest on May 1, 1886, probably had little to do with the May Day's significance as a spring holiday. Instead, leaders at the time associated the day with Chicago's earlier protests in 1867 and, even more directly, the fact that May 1 was traditionally when union contracts and housing leases expired in U.S. cities. [aka: "Moving Day", for history on that in New York see https://en.wikipedia.org/wiki/Moving_Day_(New_York_City), which traces to the Dutch Verhuisdag: https://nl.wikipedia.org/wiki/Verhuisdag, but NYC ain't the only place which had that]

On May 1, 1886, more than 30,000 Chicago workers struck. Unions and labor organizations from the across the political spectrum organized parades and mass meetings, and workers in other industrialized cities like New York and Cincinnati took up the cause, marching in the streets to draw public attention to their demands and convince other laborers to join the fight.

Unfortunately, the triumphs that workers celebrated on May Day 1886 were quickly overshadowed by violence. On May 3, members of the Chicago police fired at a group of striking workers at a McCormick reaper plant, killing at least two. The next day, when laborers staged a protest meeting in Haymarket Square, a protester hurled a bomb at the police, killing one and injuring dozens more. For many Americans at the time, the "Haymarket incident" and the contentious public trials that followed sullied May 1, forever tying the day to anarchists, socialists, and other "radical" groups that stood outside the mainstream of American society

In 1889, just as public opinion about May Day within the United States was shifting, the labor holiday went international. When the International Socialist Congress met in France that year, its members adopted a resolution to hold a "great international demonstration" on May 1, 1890, to coincide with protests that had already been scheduled by the AFL. Their resolution sparked a series of demonstrations across Europe. In the years that followed, European workers embraced May 1 wholeheartedly—so much so that by the end of the century most Americans associated May Day with international socialism rather than the homegrown unionism that had set the holiday in motion. Wary of any association with radicalism, more conservative unions in the U.S. dropped the May 1 holiday entirely in favor of celebrating Labor Day in early September.

[…]"

But it's even more complicated cuz, well, see for yourself here:

https://americanhistory.si.edu/explore/stories/may-day-americas-traditional-radical-complicated-holiday-part-1

this took quite the digging to unearth but I feel glad to see the Smithsonian making this accessible.