2 expanded a little

Simpson-semistrict categories $n$-categories could be what you're after: categories $n$-categories where everything except the unit laws holds strictly, generalising one of the crucial properties of Moore path spaces? It's not a specific definition of $n$-category, but a strictness property which can be applied within various definitions.

Carlos Simpson has conjectured that these are enough to model homotopy types; Moore path space show this in dimension one1. I know very little about the details of this myself, I'm afraid, but what I have read about it is mostly from these sources plus their links and discussions:

I believe several people have been making some progress on it recently; eg Makkai mentioned some results along these lines at the latest Octoberfest.

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Simpson-semistrict categories could be what you're after: categories where everything except the unit laws holds strictly?

Carlos Simpson has conjectured that these are enough to model homotopy types; Moore path space show this in dimension one. I know very little about the details of this myself, I'm afraid, but what I have read about it is mostly from these sources plus their links and discussions:

I believe several people have been making some progress on it recently; eg Makkai mentioned some results along these lines at the latest Octoberfest.