The "rule" you have given is a little simplistic. To use it you have to be able to write the wave solely as a function of $(kx-\omega t)$ or of $(kx + \omega t)$. That is because the thing in the brackets, the phase of the wave, has to be kept constant to apply a meaning to a direction of travel.
e.g. take $f(kx-\omega t)$. If $t$ increases, then you can only keep the phase constant by increasing $x$. So this wave travel towards positive $x$ as $t$ increases. Try experimenting with this simulation.
A standing wave cannot be written solely as $f(kx-\omega t)$ or $f(kx+\omega t)$, so is not a wave travelling in a single direction. It is the superposition of two waves of equal frequency and amplitude travelling in opposite directions.
In general, waves can always be written as the superposition of multiple waves travelling in different directions.
Your final example can be decomposed into 4 travelling waves of the same speed, 2 travelling towards positive $x$ and two towards negative $x$.
$$ y = (\sin x-t] + \sin[x+t] +\sin[2x-2t] + \sin[2x+2t])/2$$