What is the defining standard of nobility in classical history?

The proper term is comparative history. In essence when comparing across a wide spectrum with different phenomena you need to create a set of ideal types by which to do so. Technical terms therefore need to be particularly carefully chosen and weighed to see if they are applicable and represent the phenomena you are comparing.

In regard to 'nobility' I go by Timothy Reuter's definition:

Western European languages offer two other terms—nobility and aristocracy—which have been more generally used. The terms ought to be used to distinguish between two different things. A noble is, strictly speaking, a person whose (normally privileged) status is legally defined, which means that one can be a noble without exercising power. An aristocrat, by contrast, is someone who exercises power as a result of being well-born in a socially rather than legally defined sense: this implies the inheritance of wealth, power and social (but not legal) status, and it does not preclude some degree of social mobility.

Reuter, T., 'The medieval nobility in twentieth-century historiography', in The Companion to Historiography, ed. M. Bentley, pp.166-190, (London, 1997), p.167.

When I come across nobility in other periods than the Middle Ages - which is increasingly rare, I fear I'm becoming chronologically insular - I have found these definitions are applicable. The Middle Ages witnessed the exertion of power by lay aristocracy, the development of a legal milieu where nobility could emerge, and this has remained much the same for the next five hundred years. Modern republics can have aristocrats in a social sense, even when the concept of nobility is antithetical, and likewise constitutional monarchies can possess nobility without power and social aristocracies as a distinct subset or enmeshed within the nobility.

Try and locate a copy of the article if you're interested in other ways of examining the concept of nobility.