Your first option is correct.
The origin of an AoE must be the intersection of squares, and squares that are more than half covered are affected.
The area of effect of a spell, monster ability, or other feature must be translated onto squares or hexes to determine which potential targets are in the area and which aren't.
Choose an intersection of squares or hexes as the point of origin of an area of effect, then follow its rules as normal. If an area of effect is circular and covers at least half a square, it affects that square.