Is the Moon a satellite of Earth or are Earth and the Moon a twin planet system?
I would say both. Whenever a group of bodies sharing an orbit the Sun also orbit each other, the largest one is called the planet and the rest are considered satellites. So, theoretically, if you have a double planet where one component is only 1% larger than the other, we would still call them "planet" and "satellite".
Definition of a double planet is less clear. There is no obvious lower limit to how big (or massive) a satellite has to be relative to the planet in order for the system to be considered a "double planet". Pluto and Charon (1/10 of Pluto's mass) are definitely a double planet and most people would consider Earth and the Moon (1/81 of Earth's mass) a double planet.
It is possible that once some Earth-sized extrasolar planets with moons are discovered, the term "double planet" will be defined more strictly, but so far it is completely unofficial.