Your take is almost there….
I would personally highlight the object that possess the energy and the entire process of energy conversion.
At A, the person possess both GPE and KE because person is still moving and is not at its lowest point yet. (Before A, GPE is converted to mainly KE and thermal energy due to air resistance)
A to B – GPE of the person is converted to mainly EPE in elastic rope and lesser KE (since person is slowing down)
At B – Person is momentarily at rest (No KE) (before he changes direction) and at its lowest point (No GPE)… So mainly energy has been transferred to EPE in the elastic rope.
So do include the object and the energy present, which is specific.
Hence I would answer:
Person possess GPE and KE at A
At B, Person is at its lowest point and momentarily at rest (no GPE, no KE)… Elastic rope has maximum EPE.