Temperature and electric field effects on rhombohedral $(R)$ ternary ${\mathrm{Pb}(\mathrm{In}}_{1/2}{\mathrm{Nb}}_{1/2}){\mathrm{O}}_{3}\text{\ensuremath{-}}{\mathrm{Pb}(\mathrm{Mg}}_{1/3}{\mathrm{Nb}}_{2/3}{)\mathrm{O}}_{3}\text{\ensuremath{-}}{\mathrm{PbTiO}}_{3}$ (PIN-PMN-PT) ferroelectric single crystals were comprehensively studied by x-ray diffraction. We have focused on how the individual phase transitions as well as the phase transition sequences depend on thermal and electrical history. Electric field-temperature phase diagrams have been constructed under [001] field-cooling and field-heating conditions. As happens to the $R$ phase of binary PMN-PT crystals [H. Cao, J. F. Li, D. Viehland, and G. Y. Xu, Phys. Rev. B 73, 184110 (2006)], the $R$ phase of the zero-field-cooled (ZFC) state is replaced by a monoclinic A $(M$) phase in the field-cooled (FC) diagram. In particular, reciprocal-space mesh scans demonstrated that the $M$ phase was stable for crystals poled along the [001] crystallographic direction rather than the initial $R$ phase of the ZFC state. Furthermore, an $E$-field-induced phase transformational sequence of $R\ensuremath{\rightarrow}M$\ensuremath{\rightarrow} tetragonal $(T)$ was observed at constant temperature, revealing a gradual increase in the $c$ lattice parameter. These findings demonstrate that the ternary PIN-PMN-PT crystals exhibit common phase transition features with binary PMN-PT and ${\mathrm{Pb}(\mathrm{Zn}}_{1/3}{\mathrm{Nb}}_{2/3}{)\mathrm{O}}_{3}\text{\ensuremath{-}}{\mathrm{PbTiO}}_{3}$ (PZN-PT) ones for compositions in the low PT side of the morphotropic phase boundary.