These are multiple changes which probably don't belong into one pull request, but since some of them depend on each other, I've added them to just one branch.
Because of this, please don't read the changes as a whole and check each commit individually. If this gets merged, please don't squash.

The main intention of these changes was to rewrite the "Source code" section and suggest to rather use the --user parameter on Linux. The reason for this is that installing python packages system-wide via pip or setuptools can cause issues on certain Linux distros, because packages installed into /usr can conflict with packages of the distro's regular package management. Therefore we should not recommend installing streamlink system-wide. I've also removed this from the script that generates the github releases page.

In order to keep the overall install page more simple, compact, precise and most importantly, consistent, I've also removed the shell prompts from each code block and added sudo to each line where root permissions are required (previously annotated with the # shell prompt instead of $). I am not sure though if sudo is available everywhere and if it always makes sense. However, with this change, it's now possible to use comments in the code blocks, as they now can use proper highlighting.

As a last step, I have split the "other platforms" section into Windows and macOS and moved these two sections to the top. The reason for this is that people generally don't (want to) read and since these OSes are (unfortunately) more common, it's simply better to have them at the top of the document. The detailed Windows instructions are still at the bottom, which means that the old layout doesn't get changed that much. The section anchors however were renamed, which might break links from other documents or somewhere else, not sure.

As a reminder of how to build and see the docs locally:

make --directory=docs clean html
chromium ./docs/_build/html/install.html &