The second- and third-order azimuthal anisotropy Fourier harmonics of charged particles produced in $\text{pPb}$ collisions, at $\sqrt{{s}_{NN}}=8.16\phantom{\rule{0.16em}{0ex}}\text{TeV}$, are studied over a wide range of event multiplicities. Multiparticle correlations are used to isolate global properties stemming from the collision overlap geometry. The second-order ``elliptic'' harmonic moment is obtained with high precision through four-, six-, and eight-particle correlations and, for the first time, the third-order ``triangular'' harmonic moment is studied using four-particle correlations. A sample of peripheral $\text{PbPb}$ collisions at $\sqrt{{s}_{NN}}=5.02\phantom{\rule{0.16em}{0ex}}\text{TeV}$ that covers a similar range of event multiplicities as the $\text{pPb}$ results is also analyzed. Model calculations of initial-state fluctuations in $\text{pPb}$ and $\text{PbPb}$ collisions can be directly compared to the high-precision experimental results. This work provides new insight into the fluctuation-driven origin of the ${v}_{3}$ coefficients in $\text{pPb}$ and $\text{PbPb}$ collisions, and into the dominating overall collision geometry in $\text{PbPb}$ collisions at the earliest stages of heavy ion interactions.