Understanding persistence properties of a system with respect to its underlying microscopic dynamics provides important insights into the associated first-passage behavior. Here, we investigate the persistence properties during phase ordering in the two-dimensional ($d=2$) Ising model evolving under competing nonconserved spin-flip and conserved spin-exchange dynamics by means of Monte Carlo simulations at zero temperature. We examine three distinct persistence probabilities: (i) the total persistence probability, defined as the probability that a lattice site has never experienced a change in the sign of the spin residing there; (ii) the spin-flip persistence probability, which exclusively measures the fraction of sites that have never undergone a spin-flip event; and (iii) the composite persistence probability, defined as the fraction of sites that have experienced neither a spin-flip nor a spin-exchange event. In the asymptotic regime, both the total and spin-flip persistence probabilities exhibit identical power-law decay, irrespective of the relative occurrence probability of the spin-flip move $p_r$. The corresponding persistence exponent ${\theta }_i\approx 0.225$, is found to be consistent with the value reported for systems evolving purely under nonconserved dynamics. We further demonstrate that both persistence measures satisfy the scaling relation $d-d_f^i={\theta }_i/{\alpha }_i$, where $d_f^i$ is the fractal dimension of the corresponding persistence lattice and ${\alpha }_i\approx 1/2$ characterizes the asymptotic power-law growth of spatially correlated regions of non-persistent spins. In contrast, although the composite persistence probability also exhibits asymptotic power-law decay, both the corresponding persistence exponent $\theta_{\rm c}$ and the fractal dimension $d_f^{\rm c}$ of the persistence lattice show strong dependence on $p_r$. Combined with the presence of a universal growth exponent $\alpha_{\rm c}\approx 1/2$, this leads to the breakdown of the scaling relation among the characteristic exponents for the composite persistence.