How to call the two distinct points on a heart-shaped curve

You are correct in that the term for this is a cusp. (See also cardioid curve).

As far as differentiating the upper one from the lower one, well, why not upper and lower?

This is indeed called a cusp. Two senses of that definition work well at the same time: the intersection of two curves, in general; and the reflection of a curve on itself.

You are right about the difficulty of convex or concave. For purely conversational English you might try internal and external, or for the heart shape per se, upper and lower would probably be generally understood.