We report the study of plasmon-exciton coupling in nanoparticles with a metal core and an outer J-aggregate shell. Our theoretical consideration shows that in many cases of practical importance the use of the conventional model of two coupled oscillators containing a single coupling constant $g$ is oversimplified for a reliable description of plexcitonic (plasmon-exciton) coupling effects in such systems. Therefore, one needs to develop a more justified approach for correct evaluation of the plexcitonic coupling efficiency. Here we demonstrate that for core-shell nanospheres two plexcitonic coupling constants, $\gup$ and $\glow$, emerge naturally in our theory due to the electromagnetic coupling of the dipolar localized surface plasmon-polariton with two excitonic shell resonances. Such resonances are caused by the splitting of a J-aggregate peak owing to the presence of inner and outer boundaries of the organic shell. Our approach is rigorously justified for hybrid core-shell nanoparticles of a spherical shape and it is more suitable for a reliable description of light extinction spectra and for quantitative determination of plexcitonic coupling efficiency than the model of two coupled oscillators. We derive simple analytical expressions for the introduced coupling constants, $\gup$ and $\glow$. Our calculations for various plexcitonic nanoparticles with a gold or silver core and an isotropic J-aggregate shell are in good agreement with the extinction spectra obtained using exact equations. This work broadens the understanding of the physical phenomena caused by the plasmon-exciton coupling and provides a new means for the correct determination of its efficiency.