ROS 1 or 2 in ROS Lunar?

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ROS 2 is (at the moment) a completely separate project, and ROS 1 releases do not share anything (roughly) with ROS 2.

Currently supported ROS 1 releases are Indigo, Kinetic, Lunar and Melodic (see REP 3 and wiki/Distributions).

The only currently supported ROS 2 release is Ardent (see ros2/ros2 - All Releases and ros2/ros2 - Roadmap).

So to answer your questions:

I would like to know which distribution is actually using ROS2?

That would be only Ardent at the moment, and the Summer 2018 release in the near future.

I looked at the Lunar distribution website and do not see the specification of its ROS kernel.

Anything currently listed on wiki.ros.org is ROS 1 specific. So none of the currently supported ROS 1 releases use anything from ROS 2.

Edit:

I thought that ROS 1 or 2 to each distribution is like Linux kernel to its distributions.

No, that is not how things are designed. While it would be nice, that is currently not possible.

I would say that a tagged set of the whole of ROS 1 or ROS 2 are the distributions. I don't believe an analogy with the Linux kernel and the various Linux distributions can be made.

Maybe should I say Ros core?

That would be ambiguous, as ros_core is the name of one of the ROS 1 variants or metapackages defined in/by REP 142 (specifically: Specification - Metapackages - ROS Core).