Isn't power defined by: $$P=\frac{dE}{dt}$$ (?)
I see in all websites that power is the amount of work done per time. Is it represented only like this? Why?
$P=\frac{dE}{dt}$ is correct.
Work and energy are so closely related that it is very easy to get them mixed up. The relationship between work and energy is this: the increase in kinetic energy of an object is equal to the amount of work done on it. Work is specifically tied to applying a force to an object over a distance, but colloquially we tend to ignore this distinction.
In theory, power should be the derivative of energy, as you wrote, but in practice work and energy are so similar and so easy to mix up that people will say "power is work done per time" and generally be close enough to the right meaning.