Since this is the only forum accessible to me, here's the question: The term "Edit mode" gets used a lot, but there is more than one layout (arrangement of tools available) for Edit Mode, for example, the initial layout, with a long column of tools running up the left side, and the vertex, edge, and face icons up next to the "Edit mode" ID, plus those same three choices spelled out at the end of the row. OR: there is a different Edit Mode Layout without those two reference clusters for choosing a target. They are absent. Gone. So if I'm in one of those layouts, how do I get to the other, and vice-versa?
If you refer to the online manual Regions entry you will see that there are two pull out bars, called the tool bar (on the left side) and the sidebar (on the right side) of the 3D viewport:
You can use hotkeys to toggle those bars so that they are visible or not. The t key will toggle the tool bar and the n key will toggle the sidebar.
Also, when the bars are not present, there is an arrow that you can drag with the mouse to pull them out:
You can also use the arrow to drag them closed. (When the mouse hovers over the arrow it changes into a two-arrow cursor.)
Update: The difference between the toolbars in your update are because of what object is being edited, which you don't show, or what tool is being used.