I think
@Helldritch is correct. The 1e DMG mentions divine intervention on behalf of PCs more than once. The Deities & Demigods gave the chances of it and spoke in my greater detail on it. Perhaps it's not the "Basic Premise," but it's the default position for gods. Specific settings like Eberron that change that default for the setting, but not for D&D as a whole.
You mention a bunch of settings.
1. Forgotten Realms. In that setting gods meddle.
2. Dragonlance. In that setting gods meddle.
3. Greyhawk. In that setting gods meddle.
4. Mystara. In that setting gods meddle. Calling them immortals is a distinction without a difference. They're still the gods.
5. Eberron. Gods don't meddle, mainly because even if they exist they can't reach the world.
6. Dark Sun. Has no gods to meddle, but if it did, they would probably meddle.
Pretty much every setting that actually has gods that can affect the world, has gods that meddle. Gods meddle.