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The papers [5, 3, 6, 19, 20] initiated the study of open $r$-spin and openFJRW intersection theories, and related them to integrable hierarchies andmirror symmetry. This paper uses a new technique, the point insertiontechnique, developed in the prequel [36], to define new open r-spin and openFJRW intersection theories. These new constructions provide potentialcandidates for theories whose existence was conjectured before:  $\bullet$ K. Hori [23] predicted the existence of open $r$-spin theory with$\lfloor\frac{r}{2}\rfloor$ types of boundary states. The one constructed in[5, 3] has only one type of boundary state. In this work we describe$\lfloor\frac{r}{2}\rfloor$ open $r$-spin theories, labelled by$\mathfrak{h}\in\{0,\ldots,\lfloor\frac{r}{2}\rfloor-1\},$ where the$\mathfrak{h}$-th one has $\mathfrak{h}+1$ boundary states. We prove that the$\mathfrak{h}=0$ theory is equivalent to the [5, 3] construction, and calculateall intersection numbers for all these theories.  $\bullet$ In [1] K. Aleshkin and C.C.M. Liu conjectured the existence of aquintic Fermat FJRW theory. We construct such an FJRW theory, and provideevidence that this is the conjectured theory.  We also explain how the point insertion technique can be used forconstructing other open enumerative theories, satisfying the same universalrecursions.

The point insertion technique and open $r$-spin theories II: intersection theories in genus-zero