In English, genitives formed with " 's " often have a possessive meaning, while "of" may function to form a kind of "genitive" with a non-possessive meaning, e.g., compare "John's photo" and "a photo of John".
However, I have not found anything similar in Latin, except in pronouns. For example, should the expression amor Dei be interpreted as the love related to or concerning God, or the love that is possessed by God? What about the word cuius?
Well, on the one hand, it seems the two functions (which Fairbairn 2011 calls relational and belonging function) are blended into the genitive. In his book, he states about the genitive function:
Genitive Function. This word comes from the Latin word for “to beget,” and it thus indicates some kind of relationship or belonging between two subjects. More generally, it is the function of indicating to whom or to what larger group a subject belongs. In English, we can perform this function using the possessive case or the preposition “of.” For example, in the phrase “David’s book,” the possessive “David’s” indicates the one to whom the book belongs. In the phrase “Lion of the tribe of Judah,” the word “of” introduces a prepositional phrase that shows the relationship between the Messiah and the Israelite tribe from which he is descended. In the phrase “Jesus of Nazareth,” the word “of” relates Jesus to the place from which he came. In a highly inflected Indo-European language, one could fulfill this relational or belonging function by placing the word (“David,” “tribe,” or “Nazareth,” in these examples) in the genitive case.
Similarly, in his online book, Lesson 23, Fr. Foster states the genitive has two possible meanings, Genitive of Object (relational) and Genitive of Ownership (belonging), but he mentions no method to differentiate them.
So far nothing special. Let's now focus on two things.
Can we re-write the belonging function, to make it explicit or evident?
Well, there is something called the dative of possessor, or possesive dative. Collar and Daniell's book on Latin suggests (page 11), as an example of the latter, Puellae est rosa, translated litteraly as "to the girl is a rose", but which intended meaning, according to the author, is "the girl has a rose" (puella rosam habet). This is unambiguous regarding the possessive function of the dative. The other alternative, as seen above, is the plain use of the verb habere, which means, among other things, "to have", or "to own".
In other words, if you are writing in Latin, alternatives include rephrasing using the possessive dative, or go straight for an explicit use of the verb "to have/own". An example of the latter could be to rephrase, for instance, "I played John's guitar" as "I played the guitar that John has". Not very elegant, but there is no ambiguity on the belonging function used.
How to differentiate between the two functions, for instance, in amor Dei?
Well, in this 1832 book about Latin and English, the author goes through many types of what seems to be called "X governing Y". This seems to be intimately related to indicating cases of possessive genitive, for instance. Here is a quotation which includes the particular example of amor Dei:
I. The Government of Substantives.
VI. One Substantive governs another in the genitive, (when the latter Substantive signifies a different thing from the former;) as,
Amor Dei, the love of God. Lex naturæ, The Law of nature. Domus Cæsāris, The house of Cæsar, or Cæsar's house.
Obs. 1. When one substantive is governed by another in the genitive, it expresses in general the relation of property or possession, and therefore is often elegantly turned into a possessive adjective; as, Domus patris, or paterna, a father's house; Filius heri or herilis, a master's son: and among the poets, Labor Herculeus, for Herculis; Ensis Evandrius, for Evandri.
Obs. 2. When the substantive noun in the genitive signifies a person, it may be taken either in an active or a passive sense; thus, Amor Dei, the love of God, either means the love of God towards us, or our love towards him: So caritas patris, signifies either the affection of a father to his children, or theirs to him. But often the substantive can only be taken either in an active or in a passive sense; thus, Timor Dei, always implies Deus timetur; and Providentia Dei, Deus providet. So Caritas ipsius soli, affection to the very soil, Liv. ii. 1.
Thus, many times it is evident which substantive governs the other (e.g. domus Caesaris). If the dominant is in the genitive, then it is often a possessive. Yet, when the genitive is a person (in your case, Dei), ambiguity can arise. The book goes through plenty of more examples. The scope for a full answer here might be big enough to make a whole essay about Latin!
First of all, a very important note on your use of genitive and possessive (and some incorrect assertions expressed in someone else's post). We are very fortunate that Classical studies have benefitted immensely from contemporary linguistic research. The great Latinists of the past did a lot to collect data - manually! - and they tried to describe and analyze it to the best of their knowledge, for their time. However, this is 2019, not 1919, and one has to read Latin grammars of the past (replete with most interesting language minutiae) with great caution.
By tradition (and for some other reasons), we distinguish six cases in Latin, the genitive being one of the them. Theoretically, we could posit more - if we take into account semantics solely and disregard morphology.
So, we have the genitive case in Latin that expresses different semantic relationships between the head and its dependent. Possession is only one of the genitive uses. For instance, de Groot 1956 argues there are eight "regular grammatical uses of the Genitive" in Latin.
A side note: What exactly possession is is no trivial matter, either. See the first pages with lots of examples in Baldi and Nuti 2010.
If you want to consult one of the best Latin grammars of our time, I cannot recommend enough the first volume of Harm Pinkster's Oxford Latin Syntax: The Simple Clause (OUP, 2015). It contains 1430 pages and this is just the first volume (we're still waiting, patiently, for the second volume to materialize, hopefully, in the next decade).
Now to your question. How can we distinguish between the subjective genitive and the objective genitive vs. the possessive genitive, if this is indeed what you were asking about.
Let's take a look at the following example from Pinkster (p. 1040).
Non ego illam mi dotem duco esse quae dos dicitur / sed pudicitiam .../ deum metum, parentum amorem et cognatum concordiam ...
Here, we have two uses of the objective genitive, deum metum and parentum amorem, and one use of the subjective genitive, cognatum concordiam.
This use is described in linguistics as adnominal arguments of verbal nouns.
Syntactic tests:
Sermonem tuum et Pompei cognovi ex tius litteris. (Cic.)
Appositives are an exception to (2):
...vereris ne tua domus, talis et viri et civis, ...a ceteris deseratur?
Metus hostium capit Caesarem, where the ambiguity is resolved with a direct object (experiencer) (Benedetti 2013)
cf. "“Metus” quoque et “iniuria” atque alia quaedam id genus sic utroqueversum dici possunt; nam “metus hostium” recte dicitur et cum timent hostes et cum timentur" (Gellius)
However, as Pinskter correctly observes - and this answers your question -
"For most combinations of head nouns and attributive NPs the precise semantic function of the genitive attribute is wholly dependent on the meanings of the head and attribute constituents involved and the way they are related in the extralinguistic world" (p. 1000),
cf. the examples below:
... dolorem profectionis meaa reditus dignitate consoler. (Cic.) the subjective genitive
Eo, etsi scio pol is fore meum conspectum invisum hodie (Ter.) the objective genitive
Uno eodemque tempore domus mea diripiebatur, ardebat ... (Cic.) the possessive genitive
(will add more examples and syntactic tests later and might revise the post slightly sometime this weekend)
The distinction — as well as the possible ambiguity — is nicely shown in your examples of English, in which the -'s indicates ownership, or indeed possession. 'John's photo', strictly, indicates John's ownership of a photo, and is equivalent to 'a photo of John's'. On the other hand 'a photo of John' means — and really only means — that John is the subject of the photo, and not that he is its owner.
The distinction in grammar between 'possessive' and 'genitive' is rather fine, if not actually artificial. While the latter is almost invariably used in naming the grammatical case in Latin, the former is more usually reserved to physical possession or belonging. The possessive pronouns of Latin (which are in fact adjectives) are so named to indicate the fact of ownership, not of association and, in your example, cuius is not the genitive singular ('mine') of the possessive pronoun meus, mea, meum, 'my', but of a relative pronoun (other pronouns work in the same way).
An expression such amor Dei is always ambiguous without a defining context. It is avoidable by using the 'dative of possession': in this phrase amor Deo, 'love for God', or 'the love that is given to God', removes any ambiguity.
