How to use the prepositions "apud" and "chez"?
Regarding apud, I would advise against using such an obscure term. Unless it is common in the specific publication you are writing for.
Chicago Manual of Style:
Q. When can we use apud in a note?
A. Apud (Latin for “at,” “beside,” “within”) precedes the name of an author or title to indicate a source. It is used like the French word chez to mean “in the works of” or “according to”: apud Homer. It’s appropriate for those occasions when you just want to impress classics teachers or elderly readers and it doesn’t really matter whether anyone else understands.
Publications' style guides may provide rules on how one should cite an indirect reference. For example: How to Cite a Reference Within a Reference in APA Format
[Chez was added to the question after I posted the above]
I think chez would be widely understood by (native) English speakers even if they wouldn't use it often. According to Collins Dictionary:
Used Occasionally. chez is one of the 30000 most commonly used words in the Collins dictionary
With a nice little graph showing an odd peak in the mid 1700s.
Their definition is:
at the home of
with, among, or in the manner of
Example: London is busy letting its hair down but all is obviously not well chez Belmont. Edward Docx THE CALLIGRAPHER (2003)
It can seem pretentious and so is sometimes used for humorous effect or ironically.
Note that, to answer your question, we cannot define the obscurity of apud, as it itself seems unimportant to be defined here.
And, we shouldn't say that apud has become an English word. Apud is simply a Latin word that is surrounded by the words of English. Seeing that this word is defined in Latin Dictionary, not in Cambridge, M. Webster or Oxford Dictionary, the Latin Dictionary gives its meaning: next to or at.
However, chez has been defined in various English dictionaries. Merriam Webster defines it as under:
at or in the home or business place of
Origin and Etymology:
French, from Latin casa cottage
First known use:
1940
Chez has also been defined in Urban Dictionary, English Oxford Living Dictionaries and Collins Dictionary.
Usage:
They had had a superb party chez John yesterday. [=they had had a superb party at the house of John yesterday.
We live chez my best friend. [=we live at the house of my best friend]
To cap it all, we cannot compare apud with chez and chez with apud, as its meaning is somehow not interchangeable, and they differ drastically in their origin.
Chez is a French word meaning 'at the home of'. I believe 'apud' means the same in Latin. There is no single word with the same meaning in English, so we have to use phrases like 'at our house' or 'at the Smiths' home'. English speakers familiar with the French word sometimes use it in English to express the idea in one word.
Apud is not used extensively enough in English to be considered an English word. For instance, it is not included in the Oxford English Dictionary (OED) as a lemma, that is as
a form of a word that appears as an entry in a dictionary and is used to represent all the other possible forms (Cambridge dictionary, my emphasis)
The 2nd edition of the OED contains "171,476 words in current use, and 47,156 obsolete words" (OED site).
Apud is also not included in most professionally compiled dictionaries of English. For instance, OneLook shows no examples of such dictionaries containing the word. And the resources you have cited in your question are also not professionally compiled dictionaries (such as Merriam-Webster, Oxford, American Heritage, Collins, Websters, but not wiktionary). Others have stated that apud is used in formal works to mean by means of, or through; this is an extremely specialized usage: not enough, really to make it an "English word." Thus any definitions of it come from its use in native Latin.
The OED does contain a centuries old, obsolete English preposition/adverb called anent, used as far back as the year 1000, or the time of Old English. The reason I mention this is because for this English (but obsolete) word, the OED compares its various meanings with those of both apud and chez (and even the German neben), as in
In the company of, with, among, beside, by (Latin apud, French chez, German neben)
also
With (figuratively), according to the way or manner of (Latin apud)
also
On a level with in position, rank, or value; equal to, on a par with
also
In a line with, side by side with, in company with, beside
I am quoting definitions of the obsolete English word anent to show that such prepositions/adverbs can have several associated meanings (and also because I had never heard of it before and it seems a nifty word). I hope this helps explain the added concern in your question expressed by "More shockingly, I found nine meanings of apud on Word Hippo site, i.e. among, at, before, amongst, about, beside, near, in the presence of." Such words, whether in English or other languages such as Latin can indeed have a host of meanings.
But again, it is important to distinguish between professionally compiled dictionaries (such as Merriam-Webster, Oxford, American Heritage, Collins, Websters, but not wiktionary) that deal with English words and sources that might list words that are not usually considered to be English words, such as apud.
Take-home message:
Despite the presence of apud in WordHippo, it is not an English word, and the multiple meanings there are really meanings that it has in Latin, not in English. 99% of English speakers have never heard of the word.
On the other hand...
Chez, as can be seen by the definition of ament, is one French equivalent of the Latin apud, with the difference that chez, like other borrowed words such as gratis, rodeo and lasso, has been used so often by native English speakers (in this case to mean "at the house or home of" (OED)) that it is considered an English word, and it appears as a lemma in the OED and several other good English dictionaries. See OneLook, for example. There you will find definitions in Merriam-Webster, Oxford, American Heritage, Collins, Websters, all of which are what I mean by professionally compiled dictionaries. Wiktionary is not included in that category and I highly, strongly suggest you do not rely on it; in fact, I suggest you as a learner don't use it.