The video from Brian Greene is completely correct and the equations are actually the exact equations that professional physicists – or graduate students learning the physics – use.
The only "unusual" twist is a change of the notation that Brian wisely made to simplify the formulae and make them more intuitive. When he writes the total Higgs field $\Phi$, he actually means $(m_\Psi/v)\phi$ where $\phi$ is the usual physicist's normalization of the Higgs doublet field.
This $\phi$ may be rewritten as $v+h$ in the usual notation; $v=246$ GeV is the vacuum expectation value – also known as the vacuum Higgs condensate or the Higgs syrup penetrating all of space, Higgs aether, and so on – while $h$ is the oscillating quantum field measuring the variation of the Higgs field around the vacuum expectation values. When we multiply the usual formula
$$\phi = v+h$$
by $m_\Psi/v$ and choose the symbol $H = (m_\Psi/v) h$ for the rescaled variation of the Higgs field, we will get Brian's
$$ \Phi = m+H $$
and the remaining formulae follow. Incidentally, $m_\Psi/v$ is also equal and may be written as $y_\Psi$, the Yukawa coupling (interaction constant) encoding the strength of the interaction of the Higgs field $h$ with the fermionic field $\Psi$.
At any rate, it is true that from the $v+h$ decomposition substituted to the interaction term $\phi \Psi\Psi$, one gets both the mass term from the constant, $v$ part of $\phi$, as well as the usual cubic interaction $y h \Psi\Psi$.
Well, there are some other details that Brian simplifies. For example, $\Psi\Psi$ should be $\bar \Psi\Psi$, one of the terms should be barred (complex conjugate), and there should be additional indices because there are many fermionic (lepton and quark) fields in the Standard Model, and so on, but these subtleties are not important for the understanding of the basic Higgs mechanism which is why Brian suppressed them.