Lowercase "moon", "sun" and "solar system"?

From a grade school textbook:

Good morning, children. I'm an astronomer. I study the stars and the planets. They're amazing!

We live on the Earth. The Earth is a planet. It rotates all the time. The Earth takes 24 hours to rotate completely.

There are eight planets in the solar system and they all go round the sun. Some planets go quickly and some planets go slowly. The Earth takes 365 days to go round the sun.

At night you can see the moon and the stars. The moon goes round the Earth. The moon takes 28 days to go round the Earth.

In the day you can see the sun. The sun is a very big star. It's 100 bigger than the Earth! It's the only star you can see in the day.

[ Stella Maidment and Lorena Roberts, Happy Street, New Edition, Class Book 2, Oxford University Press, published 2009, 2013, 2014 ]

This work may contain some astronomical errors, but that is not the theme now. I've highlighted some instances where two bodies appear in the same sentence, including "sun" and "moon", and where only one takes a capital letter (Earth). I'm reading this Q&A (in particular this answer) on the use of definite articles with earth/moon/sun; the Wikipedia article on capitalization has a reference to the Mayfield Handbook of Technical and Scientific Writing, which suggests to:

Capitalize astronomical terms such as the names of galaxies, constellations, stars, planets and their satellites, and asteroids. However, the terms earth, sun, and moon are often not capitalized unless they appear in a sentence that refers to other astronomical bodies.

The sun is an ordinary star.
Venus and Earth differ significantly in the composition of their atmospheres.

[ Mayfield Handbook of Technical and Scientific Writing, section 9.1 ]

Is capitalization preferred with Sun, Moon and Solar system, as in the the star, the satellite, and the body containing them; is it generally accepted to give specific consideration (as does the MHoTaSW) to cases where more than one name appears in the same sentence, and to prefer lowercase when there is only one?

Solution 1:

This text has a very annoying way of talking down to children by using absurdly short sentences. It also has one blatant astronomical error ("The moon takes 28 days to go round the Earth."). But I do not think it contains any real grammatical errors, just infelicities.