As another example:
This is a lattice of lattices from section 6.3 in the book on Formal Concept Analysis from Wille and Ganter (ISBN 978-3-642-59830-2).
That is an example of a concept lattice. Another example can be [this](https://en.wikipedia.org/wiki/Magma_(algebra)#Classification_by_properties) classification in wikipedia. From the "Group-like structures" table there's a general procedure that gives the Hasse diagram of the corresponding lattice that gives the same representation of inclusions as you depict. The relation there is the predication of a fundamental property said of a class of structures (e. g. "groups require invertibility"). It's nice how the diagram emerges by magic from the table and, as you nicely say, that reflects what happens in our heads when we organize that flood of information.
This ideas cry for a categorical treatment. A formal context determines a Galois connection, hence an adjunction, and prof. Willerton has written in the [nCafe](https://golem.ph.utexas.edu/category/2013/09/formal_concept_analysis.html#more)
> Several of you will have realised ... [that] the concept lattice is the centre of the adjunction