The first image is in Normal Z direction and the second image is in Local Z direction.
If you are in Global Orientation, moving in Z (G, Z) is using the Global Z-Axis. Pressing Z again (G, Z, Z) switches to Local Z-Axis and pressing Z a third time (G, Z, Z, Z) switches to ‘no Axis; free movement.
Play with the Default Cube, Rotate it in Object Mode and then start moving it in any Axis, pressing the Axis letter several times.Use different Orientations and see what happens.
Extruding is Extrude and Move, so that is basically the same thing..it’s just the moving, that confuses you.
