Is there an established metric in historiography for what constitutes a “modernized” country?

There's no established metric. There's in fact not even a consensus definition for modernisation - perhaps because the idea of modernity is itself quite elusive.

A traditional definition is to treat modernisation as synonymous with westernisation. As Shmuel Eisenstadt describes, it is:

the process of change towards those types of social, economic and political systems that have developed in Western Europe and North America from the seventeenth to the nineteenth century and spread to other European countries and nineteenth and twentieth centuries to the South American, Asian and African countries.

_{Eisenstadt, Shmuel Noah. Modernization: Protest and Change. Pretince-Hall Inc, 1966.}

Yet some writers treat modernity as more or less equivalent to economic development. Under this meaning, rising economic output offers a possible metric of modernisation in and of itself. Levy, for example, provides a definition that seem based off productivity:

A society will be considered more or less modernised to the extent that its members use inanimate sources of power and/or use tools to multiply the effects of their efforts.

_{Levy, Marion Joseph. Modernization and the Structure of Societies: A Setting for International Affairs, Jr, Princeton University Press, 1966.}

Alternatively, if modernity equates high economic productivity, then modernisation could be considered the process of making that possible. Rostow's definition thus defines modernisation as achieving the preconditions for such an "economic take-off":

Modernisation theory is epitomised in Rostow's (1960) influential model of 'stages of economic growth'. These described how 'traditional' societies (with 'primitive' technologies and spiritual attitudes to nature), 'develop' to 'pre-conditions for economic take off' (like that experienced in seventeenth and eighteenth-century Western Europe). 'Take off' follows, where new industries and entrepreneurial classes emerge.

_{Pepper, David. Modern Environmentalism: An Introduction. Psychology Press, 1996.}

Another view is that modernisation is a relative concept. That is, the idea of Meiji Japan modernising only exists because she was surpassed by the West and wanted to eliminate the gap. In this view, modernity is not about the economy per se, but rather about adopting the necessary institutions to keep up from now on.

Black's definition is in this vein, tying modernity to the ability to partake in the scientific revolution:

the process by which historically evolved institutions are adapted to the rapidly changing functions that reflect the unprecedented increase in man's knowledge, permitting control over his environment, that accompanied the scientific revolution.

_{Black, Cyril Edwin. The Dynamics of Modernization: A Study in Comparative History. New York: Harper & Row, 1966.}

Still other writers have adopted a less concrete view of what it means to be modern. Rather than a fixed pattern of development such as that modeled after the Western experience, they argue modernity takes on many forms, and is better characterised by a set of features. Shmuel Eisenstadt argues, for instance, that modernity is about an ongoing process of transformation:

the best way to understand the contemporary world - indeed to explain the history of modernity - is to see it as a story of continual constitution and reconstitution of a multiplicity of cultural programs and cultural patterns of modernity.

_{Eisenstadt, Shmuel N. "Some observations on multiple modernities." Reflections on Multiple Modernities: European, Chinese and other Interpretations (2002): 27-41.}

Writers like Arnason takes this one step further and describes modernity in libertarian terms:

The idea that the significance placed on human agency ultimately defines modernity is shared by others. Johann Arnason has described modernity as an 'unprecedented affirmation of human autonomy'.

_{Ichijo, Atsuko, ed. Europe, nations and Modernity. Springer, 2011.}

In other words, in this view, modernisation is about creating the environment, such as a free and open political system, that would allow a society to continuously adapt and transform itself.