The answer is: both, but for different contexts.
The HR diagram is used to display the properties of observed stars. A point on the diagram represents a star with a certain effective temperature and luminosity, which can be interpreted as being indicative of the stage of development of the star. Equivalently, the diagram can be expressed in terms of absolute magnitude and stellar classification/type. The diagram was originally constructed in 1911 by E. Hertzsprung and H. N. Russell independently, though we predominantly use Russell's version nowadays.
As one simulates the theoretical evolution of a star, the star traces an evolutionary "track" on the HR diagram, which is a line that runs across the diagram in various directions representing the evolution of the star through its various development stages, as in the second figure in the OP. This is useful for displaying the effect of various parameters on the complicated evolution of stars, see here for example.
These two efforts are related historically: as better telescopes were made, there were more and more stars observed, and more and more points were added the HR diagrams, but eventually we needed computers to handle the data. Along that route, the development of theories of stellar evolution were/are intimately tied to the development of observational astronomy, as experiment and theory always are, and the HR diagram has been a powerful tool for advancing theories of stellar astrophysics.
Looking at the first image, we might think that the subgiant branch is the approximate line of stars formed by different subgiant stars on the HR diagram. While looking at the second one, we might think that the expression branch means the evolutionary track of an individual star.
So you can think of the stars in the first image of the OP as snapshots of each of the stars at a point in their lifetimes. The stars that group in the sub-giant branch (so called for low and intermediate mass stars, but is called the Hertzsprung gap for high mass stars because it's so short lived) are in the part of their evolution where hydrogen burning in their core has ceased and hydrogen shell burning has begun. The evolutionary track in the second image in the OP does indeed show the evolution of a single star through its stages of development.