The Earth/Moon system does not qualify as a binary planet because it does not meet the L4/L5 instability threshold. In a system of two orbiting masses, the larger needs to have at least 25x the mass of the smaller for the system to have stable L4/L5 points. Earth is ~80x more massive than the Moon, allowing the system to have stable L4/L5 points, and is therefore a satellite system.
The Earth/Moon system does not qualify as a binary planet because it does not meet the L4/L5 instability threshold. In a system of two orbiting masses, the larger needs to have at least 25x the mass of the smaller for the system to have stable L4/L5 points. Earth is ~80x more massive than the Moon, allowing the system to have stable L4/L5 points, and is therefore a satellite system.
Quick diagram for anyone wondering
L4 and L5 get unstable if the masses orbiting each other are too similar