Definability of 1-Types in Weakly $o$-Minimal Theories

In the article, we prove a criterion for definability of 1-types over sets in weakly $o$-minimal theories in terms of left and right convergences of a formula to a type.
Van den Dries proved that every type over the field of reals is definable. Marker and Steinhorn strengthened his result. They (and, later, Pillay) proved the following assertion. Let $M\prec N$ be a pair of models of some $o$-minimal theory. If, for each element of $N$, the type of this element over $M$ is definable then, for each tuple of elements of $N$, the type of this tuple over $M$ is definable.
We construct a weakly $o$-minimal theory for which the Marker–Steinhorn theorem fails; i. e., some pair of models of the theory possesses the following property: For all elements of the larger model, the 1-type over the smaller model is definable but there exists a tuple of elements of the larger model whose 2-type over the smaller model is not definable.