λ_R(R_e)−ε(3R_e) diagram color-coded by κ_rot(all), f_spheroid(all), f_halo(all), and f_bulge(all) for TNG50 (upper) and TNG100 (lower). Here, κ_rot(all), f_spheroid(all), f_halo(all), and f_bulge(all) represent the significance of cylindrical rotation, the spheroidal mass fraction, the stellar halo mass fraction, and the bulge mass fraction, respectively (see Sections 4.3 and 4.4). The magenta curve shows the λ_R − ε relation for edge-on, axisymmetric galaxies derived by Cappellari et al. (2007). The dotted black curves illustrate this relation across varying i, with steps of Δi = 10°. Additionally, the thin dashed black lines represent the theoretical distribution for the different rotations spaced at intervals of Δλ_R, intr = 0.1. The bold black curve represents our proposed new threshold for distinguishing between SRs and FRs. This threshold is derived from the tensor virial theorem for oblate galaxies with intrinsic parameters λ_R, intr(R_e) = 0.4 and ε_intr = 0.525, assuming an anisotropy of δ = 0.7ε_intr = 0.367. For comparison, the thin solid black line represents the standard empirical threshold defined by ${\lambda }_R(R_{\mathrm{e}})=0.31\times \sqrt{\epsilon }$ proposed in Emsellem et al. (2011, Eq. (3)). We present the proportion of SRs classified by our new threshold alongside the standard old threshold at the upper-left corner. We smoothed the color data using the locally weighted regression method LOESS (Cleveland & Devlin 1988; Cappellari et al. 2013a) with a smoothing factor of 0.1. The horizontal dashed blue line represents the empirical threshold defined by λ_R(R_e) = 0.1 according to Emsellem et al. (2007); the black polygon denotes the empirical threshold set by λ_R(R_e) < 0.08 + ε/4  and  ε < 0.4 as per Cappellari (2016); and the green polygon signifies the empirical threshold given by ${\lambda }_R(R_{\mathrm{e}})<{\lambda }_{R\mathrm{start}}+\epsilon /4\mathrm{and}\epsilon <0.35+\frac{{\lambda }_{R\mathrm{start}}}{1.538}$ (with λ_Rstart = 0.12 when the IFS data quality is comparable to that of SAMI) based on van de Sande et al. (2021).