This is a confusion caused by people using language differently. When doing mathematics, we must be quite precise in assigning meanings to different terms, which is why we take so much trouble to define them. And the point of a definition is, once we've understood what somebody means by a term, we won't make the mistake of thinking they mean something else!
So here's how we use certain common English words in ordinary ("Euclidean") geometry:
- A circle is a closed curve in space, formed by the set of all
points an equal distance from a fixed point called the centre of
- A disk (or disc) is the set of all points contained inside (*)
a circle, including its centre.
Note that a disk does not necessarily include the circle itself! (When studying topology - which is rather like geometry would be if distances weren't fixed or important - we distinguish between a closed disk, which includes the boundary or outside edge, and an open disk, which doesn't. And we often call a disk a ball - think of a squishy rubber ball.)
We also define a convex set to be one that includes every point that lies on a line segment joining any two of its points. So you can see that the inside (or interior) of a disk is a convex set, since even in a squishy rubber ball, any one point that lies between two others of the interior must be in the interior, no matter how much we squish it. But the outside (or boundary) of a disk is not a convex set, since there are points between any two points on the circle that don't also lie on the boundary, that is, they lie inside the circle, in the interior of the open disk. For example, as pointed out in the answer by celtschk, the intersection of those two chords AB and CD in your diagram is such an interior point that lies between points of the circle.
(*) Now I haven't defined what I mean by point, inside or a line segment. But in the case of a circle in Euclidean geometry, the ordinary notion of "inside" and "outside" works well enough; a point is as small a dot as we can possibly draw in a picture; and a line segment is that part of a straight line that connects two points. More generally, we can't define everything! Even in mathematics, where we try to be as precise as we can, we sometimes have to say "Enough!" and just accept certain ideas as basic, undefinable notions. We usually take a geometric point as one of those basic notions.