This is correct if $P(S)$ is not contained in the support of $\mathrm{div}(\omega)$. It comes essentially from the definition of $i_x(K_X, P)$. You don't need $\omega$ to be an exact differential from. However the intersection number depends on the choice of $\omega$ (as well as the Weil divisor $K_X$). You can check this by yourself by multiplying $\omega$ by a non-zero constant in $K(S)$ and see the effect on the total intersection number. If $P(S)$ is contained in the support of $K_X$, then you can't define $i_x(K_X, P)$.
EDIT. Let me add some more details. Denote by $K(X)$ the field of rational functions on $X$, viewed as a constant sheaf on $X$. Then $\omega\in \omega_{X/S}\otimes K(X)$. Hence $\omega\cdot\omega_{X/S}^{\vee}$ is a subsheaf of $K(X)$, hence equal (not only isomorphic) to $O_X(-D)$ for some Cartier divisor $D$ on $X$. We have $$\omega_{X/S}=\omega\cdot O_X(D).$$ A straightforward local computation shows that $\mathrm{div}(\omega)=[D]$ the Weil divisor associated to $D$. \mathrm{div}(\omega)=D$as Cartier divisors. Let us identify$P$with$P(S)$. Let$I\subset O_X$be the ideal sheaf defining$P$in$X$. As$P$is not contained in the support of$D$,$D|_P$is a well defined Cartier divisor on$P$. Namely, if a local equation of$D$at some point$x\in P$is given by$f_x\in K(X)$, then we can write$f_x=a/b$with$a, b\notin I_x$(here we use that the fact that$O_{X,x}$is a UFD). Then a local equation of$D$restricted to$P$is$\bar{a}/\bar{b}$where$\bar{c}$means the image of$c$in$O_{X,x}/I$. The above equality restricted to$P$reads $$P^{*}(\omega_{X/S})=P^{*}(\omega) \cdot O_P(D|_P).$$ So$P^{*}(\omega)$is a rational section of$P^{*}\omega_{X/S}$and its divisor on$S$is$D|_P$. To get an intersection number independent of the choice of a rational section$\omega$, you have to use Arakelov intersection theory. 2 arggg EDIT. Let me add some more details. Denote by$K(X)$the field of rational functions on$X$, viewed as a constant sheaf on$X$. Then$\omega\in \omega_{X/S}\otimes K(X)$. Hence$\omega\cdot\omega_{X/S}^{\vee}$is a subsheaf of$K(X)$, hence equal (not only isomorphic) to$O_X(-D)$for some Cartier divisor$D$on$X$. We have $$\omega_{X/S}=\omega\cdot O_X(D).$$ A straightforward local computation shows that$\mathrm{div}(\omega)=[D]$the Weil divisor associated to$D$. Let us identify$P$with$P(S)$. Let$I\subset O_X$be the ideal sheaf defining$P$in$X$. As$P$is not contained in the support of$D$,$D|_P$is a well defined Cartier divisor on$P$. Namely, if a local equation of$D$at some point$x\in P$is given by$f_x\in K(X)$, then we can write$f_x=a/b$with$a, b\notin I_x$(here we use that fact that$O_{X,x}$is a UFD). Then a local equation of$D$restricted to$P$is$\bar{a}/\bar{b}$where$\bar{c}$means the image of$c$in$O_{X,x}/I$. The above equality restricted to$P$reads$$P^{*}(\omega_{X/S})=P^{*}(\omega) \cdot O_P(D|_P).$$ So$P^{*}(\omega)$is a rational section of$P^{*}\omega_{X/S}$and its divisor on$S$is$D|_P$. 1 This is correct if$P(S)$is not contained in the support of$\mathrm{div}(\omega)$. It comes essentially from the definition of$i_x(K_X, P)$. You don't need$\omega$to be an exact differential from. However the intersection number depends on the choice of$\omega$(as well as the Weil divisor$K_X$). You can check this by yourself by multiplying$\omega$by a non-zero constant in$K(S)$and see the effect on the total intersection number. If$P(S)$is contained in the support of$K_X$, then you can't define$i_x(K_X, P)$. To get an intersection number independent of the choice of a rational section$\omega\$, you have to use Arakelov intersection theory.