Simplified equations for FM (frequency modulation) synthesis and PM (phase modulation)? Clicking on the links I find on the web it seems many authors don't express them
neatly enough for me or they're even confusing the two. Can someone please clear this up? 

Is frequency modulation 

\begin{align}
&amp;\rm  \sin((frequency+modulator)time-phase)\\
&amp;\qquad\quad\textrm{or is it}\\ 
&amp; \rm  \sin(frequency*time+modulator-phase)\quad ? 
\end{align}

Ugh!
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 $t=0$.