Xanathar's Guide to Everything has a few variant options for playing on a Grid.

XGtE, Page 86:

... Choose an intersection of squares as the point of origin of an area of effect ...

... If any part of a square is under the template, that square is included in the area of effect. ...

Diagrams 2.1 & 2.2 show this, with the Cone's origin being on the intersection. Its shadow clearly covers small sections of both squares just after the intersection.

PHB, Page 205:

A line extends from its point of origin in a straight path up to its length and covers an area defined by its width.

My interpretation is that this means using templates on grids increases the average Area of Effect size notably for Cones & Lines particularly.

Does that seem correct?

Visual example: enter image description here

Which would affect all 4 enemies labeled 1 if the caster is the one labeled 2.