Short summary: The mass of the planet determines the speed of the satellite (and of course other aspects of the satellite's orbit).
What the statement does NOT say is that more massive satellites move faster or slower or in different orbits than less massive satellites. I believe that misconception to be the source of your error.
Now when we consider the solar system in light of our (new??) understanding of the quote, we see that for the solar system the SUN takes the place in the quote of the massive planet while the PLANETS, because they orbit the SUN, take the place of the satellites.
So substitute SUN for 'massive planet' and PLANETS for 'satellites' in your quote and observe the result. In this case, there is no substitute for 'lesser mass planet' in the quote because there is no second SUN for the planets to orbit. But we can say that if we considered planets orbiting the massive star Betelguese, that the quote accurately suggests that such planets would have to move faster their orbits that would planets a similar distance from our sun Sol.
Or to put it in simpler terms, the centripetal force created by the circular orbit must equal the force of gravity that is acting in the opposite direction. Gravity increases with the mass of the star, and decreases with distance.
Just think of how your hot wheel cars are able to go around the loop and stay stuck to the track.
Edited by Taq, : No reason given.