As I understand it (although to me it is somewhat ambiguous so apologies if I have misunderstood) the answer provided by @Mari-LouA implies that if the word “earth” is not capitalized it is not a proper noun, even where it refers to the planet, as in the King James Bible (Math 24:35):
Heaven and earth shall pass away…
and in many other examples in answers to the SE question on the spelling of earth/Earth both with and without the definite article.
If so, I would disagree. It seems to me to come down to the distinction between what is a ‘proper noun’ and what are the consequences of a word being a proper noun. Let us look at a standard definition, e.g. from SE ELU tag.
A proper noun or proper name is a capitalized noun representing a unique entity as opposed to a common noun, which represents a class of entities or non-unique instances of that class.
And Lexico has a similar definition in which both the description and the orthography are part of the definition.
On this basis earth can be neither a proper noun (not capitalized) nor a common noun (represents a unique entity).
This seems to me neither proper logic nor common sense.
My answer (supported by common sense and the rather long abstract from the Wikipedia article below) is:
The earth (and sun and moon) when used to refer to the planet in common writing (rather than technical astronomical writing where they lack a determiner and are capitalized) is a proper noun because it refers to a unique entity. The fact that it is not capitalized is exceptional but merely an irregularity in language and orthography.
Is this important linguistically? I may be wrong, but I am unaware of any other reason to classify nouns as proper or common other than orthography. If I am correct (he ducks) then this is a useful help for orthography but not an infallible rule. This is English. Who is surprised?
Abstract from Wikipedia entry (with my emphasis)
A proper noun is a noun that identifies a single entity and is used to refer to that entity, such as London, Jupiter, Sarah, or Microsoft, as distinguished from a common noun, which is a noun that refers to a class of entities …The detailed definition of the term is problematic and, to an extent, governed by convention.
…A distinction is normally made in current linguistics between proper nouns and proper names. By this strict distinction, because the term noun is used for a class of single words (tree, beauty), only single-word proper names are proper nouns: Peter and Africa are both proper names and proper nouns; but Peter the Great and South Africa, while they are proper names, are not proper nouns.
…In English and many other languages, proper names and words derived from them are associated with capitalization; but the details are complex, and vary from language to language.
Coda
I suppose I should propose a change in the tag, assuming I have enough rep to do so:
A proper noun or proper name is a noun representing a unique entity as opposed to a common noun, which represents a class of entities or non-unique instances of that class. Proper nouns are generally capitalized in English, but there are a few (e.g. earth) which are not.