Employed as a telegraph clerk trying to write faster and more legibly, Edison developed a manuscript print style, and it was perpetuated among and imitated by other telegraph clerks.
While Edison's style remains largely unacknowledged in books on the history of penmanship, it clearly is upright, unlike the style of the time. However, it is fair to say that he neither authored a book on penmanship nor taught it in any institutions, and so he cannot be seen as among those who influenced the nation's youth nor the broader public.
By contrast, one can point to the next iteration of a vertical style that was widely promoted with great vigor and purported to be backed by scientific studies. Writing teachers in Europe first adopted it, and later it came to be popularized by John Jackson in England in the 1880's. While it is hard to find the exact date that Jackson's vertical style came to be known by those in the United States, there is evidence that it was not later than 1887. Here, one can see an advertisement of it in The Educational Times
The history of vertical writing, as it comes from the Europeans and John Jackson extends into Canada with Newlands and Row in as early adopters in that country; Vaile and Harison were early adopters in the US; and many others followed in the years 1894 - 1904. Afterwards, the trend reversed itself and most people abandoned the vertical style and returned to a forward slanted script--either by employing the older Spencerian script or a newer Spencerian "practical penmanship," one of the many of the assorted "practical styles," Palmer method, or other.
However, news of Edison's style, present since the late 1860s, and his efforts did, however, reach Melvil Dewey, who was perhaps inspired by Edison. No later than 1887, Dewey begins to promote "library hand," itself a vertical style, as the official style to be used by all librarians. Like Edison, he was interested in speed, efficiency, and legibility. One can think of the vertical style as the handwritten equivalent to the typewriter, which was rapidly evolving as a fast and efficient office machine that created its own vertical print.
However, likely neither Edison, Jackson, nor Dewey would have probably approved of the evolution of their vertical script ideals in the years after their innovations. For many writers, vertical script evolves into an awkward and difficult to read backhand style. Here is a library card written in approximately 1909. This particular script has the worst qualities of being somewhat ornate, which Edison had avoided; it is also not in Dewey's prescribed style. A librarian who did not follow the rules? A person who learned backhand in the era from 1894 - 1904, and whose writing had simply deteriorated like that of others?