All four curves in the OP are allowed under GR. Without knowing the context, we won't be able to answer why they were labelled like that.
The first graph shows the evolution of the scale factor with time in universes with matter+radiation (although the latter doesn't really show in such simplified representation) depending on the ratio of density vs critical density - these are labelled 'flat' and 'closed'. For the one labelled 'open' to curve upwards like that, the composition must include dark energy. If it were approaching linear expansion instead of exponential, all three would represent the possible futures of the universe as could be encountered in textbooks pre-1990s, before dark energy was seriously considered. As is, the 'open' one is a bit out of place in its company, and mislabelled, but permissible under GR nonetheless.
The graph labelled 'b) Hubble' shows the time evolution of the scale factor in universes where the only component present is dark energy. It's the only case where expansion is exponential. It is also equivalent to the Hubble parameter never changing, so it's what you get from naively assuming that the Hubble constant in the Hubble law is constant in time.
All kinds of scale factor evolutions are permissible under GR - exponential, linear, approaching steady state, contracting. Which one is appropriate depends on what's in the universe.
The graph in post #3 shows a completely different thing. That one's about how Doppler shift works.
