"Python itself" is sort of a platonic ideal. Roughly speaking it is a specification, although there is no "real" specification as there is for some other languages like C. The behavior of Python the language is defined by the documentation available on python.org. Anything that does what that documentation says it's supposed to do would "count" as being Python.
An implementation of Python is an actual program that provides that behavior. The most common one is CPython, which is what you download from python.org. The other implementations found on the "alternatives" page you mentioned are other programs that also "count as Python" in that they give the documented behavior (sometimes with some caveats), but are written independently and may, for instance, run on other platforms, run faster, run slower, do things differently under the hood, etc.
A distribution of Python is a bundle that contains an implementation of Python along with a bunch of libraries or tools. In theory, a distribution of Python could use any implementation, although all the ones I know of use CPython. The download from python.org could also be considered a distribution (a minimal distribution that doesn't contain any "extras").
You could think of it by analogy to some sort of physical machine, say an oven. "Python itself" or "Python the language" is like a description of what that machine does: it heats up, you can turn a dial to set the temperature, etc. You can't use the description to cook anything; you have to build an actual oven first. An implementation of Python would be like an actual oven that you built; as long as it does what an oven is supposed to do, it is an oven, but there could be many ways of building an oven that does the right things (wood-fired, gas-fired, solar-powered, etc.). A distribution would be an oven that comes with other things that you might often want to use along with it --- like maybe a fully-equipped kitchen that includes pots, pans, spatula, and mixing bowls as well as the oven itself.
You can indeed use all three at the same time, and in some sense you kind of have to. There is no way to use "Python the language" without using an implementation of it; it would be like saying you want to "drive a car" without driving any particular kind of car. Likewise, you can't really use an implementation without getting it as part of a distribution (at least if you consider the "bare" python.org downloads to be a distribution too).
In practice, most people using Python for practical purposes are using CPython (the de facto standard implementation), so the real choice is among different distributions of that. Among those, you mainly choose based on what extras you want to come with it. You can choose to use the "bare" distribution from python.org and then install packages yourself, or if you plan on doing scientific/analytics computing, you could choose one of the distributions geared towards that (e.g., Anaconda or Canopy).
