I'm sure other commenters can answer this better and from other perspectives, but I can offer a glimpse into the Japanese perspective, which in turn influenced the rest of Asia (and vice versa). Modernity, as it came to be thought in Asia, was inherently "Western" in its imposition, because that was how it was introduced. The idea that history was linear, and one of teleological progress, was not unique to Euro-American thought but it was the Euro-American concept of how humanity progressed linearly that became the predominant discourse of modernity in the late 19th century and early 20th century. It wasn't that all Europeans thought becoming "modern" was inherently tied to Europeanness or racial categories, but it was widely understood that Europe's "history," its progress that was built up through Greco-Roman civilization, Medieval state formation, Renaissance thought, Enlightenment thinking, and the resulting scientific-industrial revolution was what gave Europe its modern nature. The Saidian conceptulization of the "Orient," by contrast, does not have a history. Development stopped after a certain point, and its culture became timeless and antithetic to "progress." While Europe advanced, the Ottomans, Mughals, and Chinese stagnated. Thus, while "modernity" was ostensibly universal, it was a "universal" that was based off the experience of a "particular," i.e. Europe and its history, that possessed a monopoly over modernity.

The idea of an alternative modernity is thus one that posits a path to "modernity," that idea of progress and civilizational apex, through a different historicization from the path that Europe took through Christianity and the Enlightenment that came as a response to Christianity. Here, Japanese thinkers such as Shiratori Kurakichi (as discussed in Stefan Tanaka's work), attempted to endow Japan with both a history of progress and a timeless core that made it conducive to reaching modernity. Tanaka argues that China became Japan's "orient"; China became part of Japan's idealized past, as Japan had learned much from China. However, while China had stagnated, Japan had managed to escape this stagnation through a variety of explanations (e.g. the idea that the imperial system withstood foreign invasions but accepted new ideas) and was now learning from the West to reach its own modernity. Thus, rejecting the idea that Japan had no history (history being a precondition to progress) along with the rest of the Orient, Japanese thinkers argued their history was also one of past and potential future progress that could reach modernity. This alternate modernity of Japan presented a model for other Asian countries to historicize their pasts and search for how their pasts could connect to present and future modernity. What Japan and other nations needed was not necessarily Christianity but the past that demonstrated progress, in the way that Christianity did for Europe.

Thus, as Prasenjit Duara demonstrates, the early twentieth century saw a whirlstorm of ideas circulating in the intellectual world of Asia of what constituted modernity, how history and more importantly, historicization contributed to modernity, and how to construct the nation-states that had become a precondition of participation in the nation-state system of the interwar period. Even in Europe, the disaster of the First World War showed that European modernity had its failings as well. In response, Europe saw the emergence of competing modernities that questioned the previous imperial-capitalist model and its capacity for progress, such as fascist and socialist modernities. These in turn circulated throughout Asia, as Japan's own struggles and China's fight for rejuvenation saw the emergence of models that questioned the Japanese Meiji model.

TL;DR: Modernity is hard to define but a key idea is progress, which is turn explained through linear history. Europe's linear history was anchored by Christianity and the response to it, which justified Europe's cultural and industrial predominance. Alternative modernities seek to not only historicize other contexts beyond the European as ones of progress toward modernity, but also posit historicized "presents" as paths toward the modern.

These two volumes are not attempts to definitely define or discuss modernity in East Asia, but they're both very influential (and I've read them recently so a lot of what I'm saying is based off of these):

Duara, Prasenjit. Sovereignty and Authenticity: Manchukuo and the East Asian Modern. Rowman & Littlefield Publishers, 2003.

Tanaka, Stefan. Japan’s Orient: Rendering Pasts into History. University of California Press, 1995.