Title text:
Unstoppable force-carrying particles can't interact with immovable matter by definition.
Transcript:
[An arrow pointing to the right and a trapezoid are labeled as 'Unstoppable Force' and 'Immovable Object' respectively.]
[The arrow is shown as entering the trapezoid from the left and the part of it in said trapezoid is coloured gray.]
[The arrow is shown as leaving the trapezoid to the right and is coloured black.]
[Caption below the panel:] I don't see why people find this scenario to be tricky.
Source: https://xkcd.com/3084/
Not relative to the sun, relative to momentum. Changes in the magnitude or direction of velocity are objective, not relative. These translate to real changes in momentum, from any reference frame. A real change in momentum is imparted upon the Earth roughly equal to your velocity relative to the earth multiplied by your mass at the moment your contact with the Earth ceases.
ETA: I do actually agree with your salient point above: that lifting an object is relative to a given "down", and so it is meaningless to expect to be able to "lift" the most massive object in the universe.