  Flow coefficients ($v_2$ and $v_3$) are measured in high-multiplicity $p$+Au,
$d$+Au, and $^{3}$He$+$Au collisions at a center-of-mass energy of
$\sqrt{s_{_{\mathrm{NN}}}}$ = 200 GeV using the STAR detector. The measurements
are conducted using two-particle correlations with a pseudorapidity requirement
of $|\eta| <$ 0.9 and a pair gap of $|\Delta\eta|>1.0$. The primary focus of
this paper is on the analysis procedures and methods employed, especially the
subtraction of non-flow contributions. Four well-established non-flow
subtraction methods are applied to determine $v_n$, and their validity is
verified using the HIJING event generator. The $v_n$ values are compared across
the three collision systems at similar multiplicities, which allows for
cancellation of final state effects and isolation of the impact of the initial
geometry. While the $v_2$ values display differences among these collision
systems, the $v_3$ values are largely similar, consistent with the expectations
of subnucleon fluctuations in the initial geometry. The ordering of $v_n$
differs quantitatively from previous measurements obtained using two-particle
correlations with a larger rapidity gap; this difference could be partially
attributed to the effects of flow decorrelations in the rapidity direction.
