To help with the "proper definition" part of your question; you have looked at https://en.wikipedia.org/wiki/Convex_hull and gotten what could well be considered a "proper" definition but found it lacking, so perhaps a more "useful" definition

For every point inside a convex hull, a straight line to any point not within the hull will only intersect the hull once.

This is useful because given a point you can construct a line through it and test for that constructed line intersecting segments of the hull.

• No intersection the point is not in the hull.
• One intersection the point is on the hull.
• Two intersections the point is within the hull
• A straight line cannot intersect a convex hull more than twice

To help with the "proper definition" part of your question; you may have looked at https://en.wikipedia.org/wiki/Convex_hull and gotten what could well be considered a "proper" definition, but found it lacking, so perhaps a more "useful" definition is:

For every point inside a convex hull, a straight line to any point not within the hull will only intersect the hull once.

This is useful because given a point you can construct a line through it and test for that constructed line intersecting segments of the hull.

• No intersection the point is not in the hull.
• One intersection the point is on the hull.
• Two intersections the point is within the hull
• A straight line cannot intersect a convex hull more than twice