Is $E=mc^2$ false?

In the early days of special relativity it was noted that the mass of an object appeared to increase as the speed of the object approached the speed of light. It was common to see the notation $m_0$ used for the rest mass and $m$ for the relativistic mass. In this sense the equation $E = mc^2$ is always true.

However the concept of relativistic mass is deprecated these days, principally because it has caused confusion to generations of physics students. These days we use only the symbol $m$, and it always refers to rest mass. So $E = mc^2$ is only true in the rest frame of the object, and we now use the equation you mention. Having said this, in everyday life speeds tend to be low compared to the speed of light. For example if you're calculating the energy released by a nuclear reactor then using $E = mc^2$, where $m$ is the mass deficit, is an excellent approximation.

As you say, the equation:

$$ E^2 = p^2c^2 + m^2c^4 $$

works for photons as well, while the old equation did not.