The orbit hasn't changed (except through a different Milankovich cycle that I believe takes 112,000 years or something like that). What you may be thinking of its that Earth's orbital eccentricity changes on this ~100,000 year timescale.

When it comes to precession, what is happening is that the Earth's rotation axis is spinning around in space pointing in different directions on a 26,000 year cycle. This drift of Earth's spin axis is not related to its orbit.

Right now, the Northern Hemisphere (NH) spin axis is oriented toward the Sun in June. This means on the June Solstice the spin axis is most directed toward the Sun from the point of view of the NH. This is why the NH experiences summer in June.

Over the next 13,000 years, Earth's rotational axis will revolve (precess) to the other direction in space, and be oriented about 23.5 degrees in the other direction in June. This means that when Earth orbits to the December position, then the axis will orient the NH toward the Sun. So the date on the calendar of the seasons will be effectively flipped. That's it. There's nothing about the orbit that changes from precession. The seasons will drift through the calendric dates of the calendar though.

Finally, since the Earth's orbit is eccentric, and the Earth is physically closer to the Sun by 3% in January, this means that if December in 13000 years is Summer for the NH, then the NH will also get 3% more sunlight in summer.

The 13,000 year "Earth Axial Tilt Reversed" is wrong.

While the Earth precesses on a 26,000 year time scale, it isn't accurate to describe its tilt as "reversed." In 13k years, the axis will have precessed the other way, but maintain roughly the same tilt (depending on other Milankovich Cycles), as it will continue to do for tens of thousands of years after that. But the description that the N. Hemisphere is tilted toward the Sun in June is also misleading because that already happens.

What they're trying to say is that since the Earth gets closer and further from the Sun due to the eccentricity of its orbit, the Earth would be closer to the Sun while the Northern Hemisphere is tilted toward the Sun in about 14,000 years from now. This is a very small effect (about 3-4%) and hardly makes a difference in the Hemispherical temperature. The S. Hemisphere is currently closest to the Sun during their summer months (December-March) and they don't really notice. True, though, that the N. Hemisphere has more land, and would have a more profound effect. But this effect wouldn't happen all at once.

In fact, the closest the Earth gets to the Sun is in early January, so the date being "June" is also wrong unless we shift the calendar instead of shifting the seasons. It's more likely that January would be synonymous with N. Hemisphere summer, not June. And it would be about 14k years from now, not 13k.