In this answer, I only sum up information from Caspar David Friedrich (link to a great video tutorial) and Jaroslav Jerryno Novotny (application of some principles which can be found in the video too). I have upvoted both answers, however I'd like to provide a step-by-step explanation with pre-requisites to prevent learners from being blocked at some point. This is a kind of answer I was waiting for.
1. How to create a true displacement with the Material Output node
Prerequisite: This procedure is valid for Blender 2.79b.
A Material node has an input (aka socket) for a Displacement function (one of three shading components of a material object):
Three inputs of a Material Output node: Surface, Volume, Displacement
What I learned is, without specific user action, in spite of its name, the Displacement input works as the bump mapping obtained by providing a function to the Normal input of a shader (except it applies to all shaders at once):
Normal input of a shader can be used for bump mapping, not for displacement
This is the reason why the two spheres have the same aspect for the pink texture in the question.
Blender added an option to Cycles rendering engine (it seems it was with version 2.78), to customize the Displacement mode of a material. Displacement can now act as a bump mapping effect (no actual displacement of the vertices), or true displacement, or a mix of both.
However this possibility relies on features which need to be activated in Cycles: Micro-displacement, itself taking advantage of adaptive division, all that is well explained in the video linked by Caspar David Friedrich (here). These features are considered experimental in 2.79, which means they are not available by default. I don't know if activating this extended feature set permanently has side-effects, nothing related in the documentation.
Prerequisite: Activate Experimental Feature set in Render tab.
Experimental Feature Set is allowed in the Render tab of the Properties Editor
Now the displacement mode in the Material tab can be set to True displacement (as explained by Jaroslav):
Displacement modes Bump, True and Both can be set in the material tab of the Properties Editor
In the rendering below, the middle sphere displacement mode has been set to Bump. This is equivalent to the material behavior when experimental features are inactive. For the right sphere, this mode has been set to True. The material node is the same for both spheres otherwise:
A third sphere with True displacement has been added
In the third sphere, the vertices are really displaced. The level of detail is limited by the mesh grid resolution, which could be increased with a Subsurf modifier (none used here), or selecting Both for the displacement mode, which adds bump mapping to the actual displacement.
2. Displacement and Vector Displacement nodes
Nothing like this in 2.79b, even with Experimental rendering features set in (Cycles) rendering tab. Just a confusing documentation.
It seems those nodes are for the future (and are possibly available in previews / daily builds).
FWIW, I've a general comment (ok, it's more a ranting) about the official documentation and answers to questions asked by learners: If we want to allow learners to not give up with Blender, we need to be didactic and not forget about the pre-requisites when explanations are provided. Yes it's tedious, very boring, but without them, explanations can be a loss of time, they generate frustration, people cannot move forwards, and at some point some switch off to another 3D product more easy to learn (this is especially true for Blender with its, well... creative GUI). I've used Lightwave and 3ds Max a lot in the past. Tools are just tools, they don't prevent artists to create something, even if they miss some features. But without an easy way to master it, a tool can be a problem for creativity. It often boils down to availability of good learning material vs poor / confusing documentation. I just hope we, as Blender users, especially with the help of this Q&A dedicated site, can do as well as commercial products, and level off confusing explanations, including in the official documentation when this is the case, in favor of value-adding material.