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Is there a clear cut between them?

How long and how bright will the moon be for a dark night?

I googled but could not find a definition for them. People just talk about them empirically?

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What a nightmare - there is no accepted definition.

A common method is to use the fraction of the night (between the times of astronomical twilight) that the moon spends above the horizon. The problem is that this does not correlate perfectly with fractional lunar illumination or sky brightness. It does provide a good method of deciding when a night is "dark", but you get a bit of a mixed bag of "grey" nights.

It is found that the sky brightness is pretty much determined by fractional lunar illumination (FLI - the fraction of the moon's visible hemisphere illuminated by the Sun) once it exceeds about 0.15; so this is the definition that is currently used at the Isaac Newton Group of Telescopes.

Dark $0\leq$ FLI $<0.25$; Grey $0.25\leq$ FLI $<0.65$; Bright $0.65 \leq$ FLI $\leq 1$, where the FLI is calculated at 0h UT (close to midnight at the ING).

See http://www.ing.iac.es/PR/newsletter/news6/tel1.html for further discussion (in which I only just noticed I got a mention in the acknowledgements ;) ).

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