Circles and crossing planar compact convex sets

Let K-0 be a compact convex subset of the plane R-2, and assume that whenever K-1 subset of R-2 is congruent to K-0, then K-0 and K-1 are not crossing in a natural sense due to L. Fejes-Toth. A theorem of L. Fejes-Toth from 1967 states that the assumption above holds for K-0 if and only if K-0 is a disk. In a paper that appeared in 2017, the present author introduced a new concept of crossing, and proved that L. Fejes-Toth's theorem remains true if the old concept is replaced by the new one. Our purpose is to describe the hierarchy among several variants of the new concepts and the old concept of crossing. In particular, we prove that each variant of the new concept of crossing is more restrictive than the old one. Therefore, L. Fejes-Toth's theorem from 1967 becomes an immediate consequence of the 2017 characterization of circles but not conversely. Finally, a mini-survey shows that this purely geometric paper has precursors in combinatorics and, mainly, in lattice theory.