Looks like 2x6 joist lumber is not an option per IRC, and for 8' joist span your 8' beam would need to be 2-2x10s minimum (or other options depending on wood species).
IRC deck framing guide courtesy of prior question by @psaxton
The second on is a better choice; in the first one, you have a lot of angles that will make it hard to carry the loads effectively. I would probably go with a 2x6 for the platform on the second deck to get your shape; that will get you 5.5" of difference in height. You could also go with a 2x8 for a 7.5" difference. 2x4 only gets you 2x4, and 3.5" is too short to meet code for a step.
It would be good to see the beams to comment on the overall structure.
Think about lighting for the edge so that you can safely see the step at night.
This is one of the reasons free standing decks are not a good alternative to a deck that has been engineered, permitted and structurally connected to the house.
Using the house to fix what is, essentially, a foundation problem with the deck will not necessarily go well. It is not likely that all parts of the deck moved away from the house equally (structures don't usually "slide" around.) It is more likely one side has dropped opening a gap on the house side. This is common, and can usually be fixed by jacking the outside edge back to its original position.
Worse case you will need dismantle and rebuild the deck, sitting it on proper foundations constructed below the front line, and anchoring it to the home's rim joint.
Best Answer
Check out this residential wood deck construction guide.
Looks like 2x6 joist lumber is not an option per IRC, and for 8' joist span your 8' beam would need to be 2-2x10s minimum (or other options depending on wood species).
IRC deck framing guide courtesy of prior question by @psaxton