Simplified equations for FM (frequency modulation) synthesis and PM (phase modulation)?
If $t$ is time, $s(t)$ is the (appropriately scaled) signal, $\omega_0$ is the angular frequency, and $\phi_0$ is a phase offset, then phase modulation is
$$ t \mapsto \sin(\omega_0 t + \phi_0 + s(t)) $$
Different signs for $s(t)$ and/or $\phi_0$ may be used, depending on conventions and context.

Frequency modulation is more complex to write down, because we want $s(t)$ to vary the derivative of the argument to the sine as a function of time. We get something like:
$$ t \mapsto \sin\bigl(\phi_0 + {\textstyle\int_0^t(\omega_0+s(u))du} \bigr) $$
which is the same as
$$ t \mapsto \sin\bigl(\omega_0t + \phi_0 + {\textstyle\int_0^t s(u)du} \bigr) $$

We see that phase modulation is the same as frequency modulation by the derivative of the signal -- which looks like a boost of high frequencies and attenuation of low frequencies.



Your proposed
$$ t \mapsto \sin\bigl( (\omega_0+s(t))t + \phi_0 \bigr) $$
won't work -- it would look like phase modulation, with a modulation strength that increases monotonically and without bound depending on how long it is since $t=0$.