Intersection doesn't grant the basic Nocturnal shield; it grants an orange version of the shield (Nocturnal Intersection). The tooltip says the effect "does not stack"; when an ability says this, it typically refers to not stacking with itself. That is, two Diurnal ASTs couldn't both use Intersection. It's just another time to stress that two ASTs should be in opposite sects. Since it does not specifically mention incompatibility with SCH/Galvanize, at this point we should assume an Intersection shield stacks with Galvanize.
Diurnal Sect => Neutral Sect => Aspected Helios applies Nocturnal Balance; this is another specialized shield buff made for Neutral Sect.
So in all, you can have Nocturnal Field (NoctAST base shield), Nocturnal Intersection (DiAST Intersection shield) and Nocturnal Balance (DiAST Neutral Sect shield).
Just a minor correction here, Succor and Adloquium apply the same shield, Galvanize. This is the shield that's incompatible with Nocturnal Field.
Adloquium also applies a second shield on crit, refered to as Catalyze in the tooltip but appearing as "Adloquium" in test footage. The tooltip for Adloquium says, "Effect cannot be stacked with Astrologian's Nocturnal Field," however, it appears this is meant to refer specifically to the Galvanize status, as Astrologian's tooltips only mention Galvanize.
So paired with a SCH, a target should be able to hold (Galvanize or Nocturnal Field), Nocturnal Balance, Nocturnal Intersection, and Catalyze.