Learn English – No more Hiroshimas – Can proper nouns be plural form

These are all cases where the idea of having more (or less) than one Earth is considered. The other ones are of course hypothetical, but the indefinite article is used to indicate that there are other such hypothetical earths.

So, intuitively, you can consider this as 'an element from the set of hypothetical Earth-like planets' instead of the strict sense of a proper noun (although, of course, it still is a proper noun, but used as if it were a normal noun).

The same story could of course go for any other proper noun for which hypothetical alternatives could be envisioned. For example, one could write about 'a United States that never became independent of the British Empire'.

Your daughter's textbook is not quite accurate.

You capitalize such nouns when they act as names, as the ordinary form of address. "Mommy" is used that way in your second example.

But in your first example, sister is not a name, as its use with a determiner makes clear: it is a common noun denoting a relationship.

If the determiner is omitted, however, the noun is treated as a name, and it is proper to capitalize:

Ask Sister. She's fond of playing on it.

That sort of use of sister is rare, although not unknown; as Damien H says, Sister, capitalized, is usually the title (of reference and address) for a Catholic nun.