"right" uniformity condition for Nick's Class DLOGTIME is defined at http://en.wikipedia.org/wiki/DLOGTIME

$\operatorname{L}$ is defined at http://en.wikipedia.org/wiki/L_%28complexity%29

$\operatorname{NC}$ and $\operatorname{NC}^n$ are defined at http://en.wikipedia.org/wiki/NC_%28complexity%29

DLOGTIME seems to be the smallest that might work.

I've read in various places that $\, \operatorname{L} \subseteq \operatorname{NC}^2 \,$, $\,$ although every place I've

found that results which states a uniformity condition uses $\operatorname{L}$-uniformity.

Is there any deterministic class $X$ such that $\, \operatorname{L} \subseteq \operatorname{NC} \,$ is known with $X$-uniform $\operatorname{NC}$, and

1. $\;$ ... $\; X\subset \operatorname{L} \;$ is known to hold?

2. $\;$ ... $\; X\subseteq \operatorname{L} \;$ is known to hold and $\, X = \operatorname{L} \,$ is not known to hold?

(1, or to a much lesser extent 2, would seem to imply that $\operatorname{L}$-uniformity is the correct condition)
You can use $\mathsf{DLogTime}$ for uniformity of $\mathsf{NC}$ and $\mathsf{NC^2}$. There is no problem and the classes uniform $\mathsf{NC^k}$ remain the same and equal to $\mathsf{ATimeSpace}(O(\lg^k n),O(\lg n))$ (for $k\geq1$).

Generally, the only case that we need to be more careful is $\mathsf{NC^1}$ case where one should be careful about what needs to be in decidable in $\mathsf{DLogTime}$. If you use extended connection language description of circuits then everything works even in $\mathsf{NC^1}$ case.

For more on uniformity see:

Walter L. Ruzzo, "On Uniform Circuit Complexity", Journal of Computer and System Sciences, vol. 22 (1981), pp. 365–383.