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.