it's dubbed a "moon" because it's tagged to our planet's gravitational tides like Our Moon
. Yet, 2004 GU9 is a "quasi"
satellite because its orbit is dictated by other forces as well, making it unstable
This is wrong. A quasi-moon or quasi-satellite is "quasi" not because of the instability of its orbit, but because it is
not gravitationally bound (like real moons). Instead, it has an orbit around the Sun that has the same period as the Earth's, but a different eccentricity, and is roughly in phase with Earth (it is lined up with the Sun and Earth at its furthest and closest points along its orbit). When viewed from the Earth, relative to the Sun, such an asteroid appears to orbit the Earth, hence quasi-moon.
What's true is that perturbations from other forces (gravitational perturbation of other planets or asteroids during close approaches, non-gravitational effects of solar radiation) can send quasi-moons into other solar orbits (usually a horseshoe orbit). But it's not necessary. The orbit of a quasi-moon in the Earth-Sun rotating coordinate system is not a closed loop but wobbles around, while transfers between quasi-moon and horseshoe orbits are possible at specific regions of the same coordinate system, so it may just be a question of time until the quasi-moon gets to the transfer zone even without any outside perturbation.
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