Pretty much correct. To me, "linux" is just the kernel. But it is pretty common to refer to entire distributions as linux. That is what annoys RMS so much. He maintains it should be called GNU/Linux, as he sees distributions as the linux kernel plus the additional software from the GNU project. This makes sense too but I never use the term GNU/Linux. I am either talking about the kernel linux, or "linux distributions", or a specific distribution.
So yes. A distribution is just the kernel (which may include distribution specific patches) plus all the extra programs that make it usable.
The kernel is a central project, and is nominally the same in each distro, but most distros customize it a bit.
And the extra software doesn't just make the kernel more user friendly, it makes it usable at all. A kernel is just interrupt handlers, device drivers, and system calls. It basically virtualizes the hardware and provides a standard environment for programs to work on.
As far as the phrase "operating system" goes, it can be confusing. Some people may say the kernel IS the operating system, and everything else is either a utility or an application or something else. Other people may say the kernel plus some other packages make up the operating system, but most of the software is not part of the operating system. Others may say all the software in the distro forms part of the operating system.