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.
- The possessive genitive can be coordinated with a possessive adjective:
Sermonem tuum et Pompei cognovi ex tius litteris. (Cic.)
- A possessive adjective cannot be co-referential with the other noun in the genitive in such a close proximity (I've tried to avoid linguistics jargon in my answer as much as I could). This is well known from the binding theory.
Appositives are an exception to (2):
...vereris ne tua domus, talis et viri et civis, ...a ceteris deseratur?
- A canonical example is:
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)