The above is just referring to the fact that the standard "feather vs. bowling ball" question assumes the earth/moon/ground is immovable. In that case, Newton says they fall the same.

The fact that the ground is not immovable is what's being referenced

in this picture, things don't "fall," they are each accelerated towards each other.

The acceleration from gravity would be the same no matter the object mass (~9.8m/s²).

Ah but the earth doesn't just attract the ball or feather. The bowling ball attracts the earth as well, and since it has more mass, it will pull the earth towards it faster than the feather.

But if you drop them at the same time, that's moot.

So why does the bowling ball fall faster in a vacuum? Does it appear faster locally because the heavier object makes local time slower than the lighter object compared to a distant observer? I'm trying to understand what the meme is getting at.

But how would that make the bowling ball fall faster? F = G × m₁ × m₂ / r² and F = m₁ × a ⇒ a = F / m = G × m₂ / r², where m₁ is the mass of the ball and m₂ the mass of the planet. So the gravitational acceleration of a bowling ball is independent of its mass (assuming the planet has way more mass than a bowling ball).

And is the Sun was replaced with a black hole of the same mass, the Earth would just keep on rotating around it without issues, if slightly frozen

Just to add some formality to this, the original commenter might want to look up the shell theorem for classical mechanics and Birkhoff's theorem for general relativity.

Heavy is a subjective term based on the force of gravity. You are heavier if we weigh you on the earth compared to if you are weighed on the moon.

Your mass in those two examples is unchanged. The amount of mass you have is finite and not subjective like weight.