There is a missing step in the consequence you draw from the video.
In the video, the general solution of the wave equation was written as $$
\psi(x,t)= A f(x-vt) + B g(x+vt),
$$
where $f(x-vt)$ and $g(x+vt)$ represent respectively the rigid shift towards right, by a distance $vt$, of the function $f(x)$, and the rigid shift towards left, by a distance $vt$, of the function $g(x)$.
Notice that $\psi$ may represent many different physical quantities evolving in a wave-like, and $f$ and $g$ are generic functions of their arguments. Therefore, just by dimensional analysis, we can conclude that the dimensioned arguments $(x+vt)$ and $(x-vt)$ of the two functions $f$ and $g$ should appear in a dimensionless form $k(x+vt)$, where $k$ should be a quantity with the physical dimension length$^{-1}$.
In the case of sinusoidal or cosinusoidal profiles for $f$ and $g$, it is natural to use a form for $k$ written in a form that makes explicit the space periodicity over a distance $\lambda$ (the wavelength):
$$
k=\frac{2 \pi}{\lambda}.
$$
Thus, for a sin wave propagating toward the right, we have
$$
f(x-vt)=\sin \left( k(x-vt) \right) = \sin \left( kx-\omega t \right),
$$
by introducing $\omega=k v$. Similarly, for a $\cos$ wave or propagation towards the left. Then it is clear that there is no contradiction between the video and textbook, and the key point is that $\omega$ is not the velocity of the wave.