Jokes aside : being right for the wrong reasons is being wrong

I mean that's literally the point the image is trying to make. The last sentence says the point is outside the sun for Jupiter.

I don't think nitpicking the title achieves anything and it's not even misleading unless it's only taken in isolation.

Orbiting a point within the sun is still orbiting the sun.

For most bodies the barycenter, while not the same as the center of mass, is still inside the sun. This one isn't, making it notable

You're not wrong. Everything orbits the center of mass of the system, meaning the mass of the star and the body in orbit. And that is handy for astronomers, many exoplanets have been found using the Doppler spectroscopy method. Doppler spectroscopy measures the Doppler shift in the star's light as it is pulled towards and away from us by planets in orbit. The newest spectrographs are sensitive enough to detect a star's wobble caused by an Earth sized body in orbit. The barycenter is still within the star, but not at the center of the star's mass.

No, this is actually really relevant. This is part of the logic applied to labeling Pluto a dwarf planet. Pluto and it's moon do this, Earth and our moon do not. Yes, obviously the center of mass of the two isn't the exact center of the earth but it's still within the earth.

Fun fact: You actually pull the Earth up with the same force it pulls you down.. Newton’s Third Law.

Well, while we are being 'that guy', factoid is one of those words which has changed its meaning by being used wrongly for so long that the original meaning has all but vanished.

A factoid is technically supposed to be something resembling fact, but not actual fact. (The Greek suffix '-oid' normally being used for that purpose, like in paranoid, "like knowledge" or asteroid, "like a star").

The best thing about factoid, is that factoid is now a factoid. Because it resembles what it is not lol...

Anyway, nowadays, you are allowed to use it the way you did, at least in the descriptivist world view. The prescriptivists may disagree, however. And those people are often 'that guy' ;)

I'm kinda stunned that it's EVER outside the sun.

Technically speaking, no celestial body in our solar system orbits around a single point. The barycenter thing only works with two bodies. When there are more than two bodies, such as in our solar system, the orbits become chaotic. Granted, the influence between planets is small, so they all appear to orbit their barycenters with the sun, but there are small perturbations to the orbits caused by the locations and masses of all the other bodies in the solar system.

All the solar system matter contributes to an object's orbital center but that's constantly moving as the system moves.

I think (?) most planets have their barycenter inside the sun's surface

The gravitational pull of system matter pales in comparison to the sun so you don't need to consider it for amateur purposes.

You can try KSP (Vanilla) versus Kopernicus mod if you want to feel the difference.

Also called n-body

The barycenter is different for each planet-sun (or any two object) pairing.

The earth and moon have a barycenter which is beneath the surface of earth. Likewise, the barycenter of the sun-earth pair is below the surface of the sun

Edit:

The barycenter of our solar system orbits the center of our galaxy (again in a barycentric manner)

I guess they all orbit around the solar system's center of mass (negligibly affected by the universal CoM), but that CoM probably moves around as the planets themselves move.

Relative to what, you might ask? That depends who you're asking 😉

Correct.

I also believe that one of the criteria for a binary planet is that the barycenter is outside either body. Like Pluto/Charon.

Here’s an animation

Universe Sandbox

Well, mostly Jupiter and a little bit of Saturn.

No, it is not true in general that the barycenter lies outside both objects.

"outside the objects"

Gravity always wins