I agree with others that you are interpreting this correctly. The way I would articulate these concepts is that, when the genitive form of a noun version of a verb (i.e. some sort of action, but in noun form) used to modify another noun, the genitive is "subjective" if the noun being modified would be the subject of said verb, whereas it is "objective" if the noun being modified would be the object of said verb. Basically "subjective/objective" is referring to the relationship of the noun modified by the genitive, to the (verb form of the) action described in the genitive case. This is a bit abstract so I understand how it's hard to wrap your mind around it.
I also think what you've stumbled upon is that some (not all) Indo-European languages have an ambiguity or overlap in meaning between certain forms of genitive, or prepositions that originated in the genitive, and two different scenarios.
In English, some ways of wording statements are ambiguous in this way, but others are not. If you say "the love of your mother" it is ambiguous, like it could mean that your mother is either the subject or object of the love, and for this reason that phrase sounds stilted or unnatural to me as a native speaker. When the context makes it clear (such as "fear of snakes") then such constructions feel natural, as the context makes them unambiguous. And for this reason, if you wanted to talk about snakes feeling fear, you might need to use more specific wording, such as "fear felt by snakes" or "fear experienced by snakes".
If you use the simple possessive, however, with an "'s", it is also unambiguous. I.e. you hear people say "A mother's love" and it always means that the mother is feeling love, not that she is loved. Similarly, you can use the preposition "for" to signal the "objective" role here, i.e. "love for mother", but this doesn't necessarily work in all cases, like you can't say "fear for snakes" because this would mean that you would be afraid that something bad was going to happen to the snakes. For instance I might say "I fear for the snakes that cross this road."
Other Indo-European languages do not necessarily handle these things the same way. For example, in Spanish, you use different prepositions, "a" and "de", and "de" which corresponds to the genitive in Latin, is (I think only) used in the "subjective" sense. Interestingly though, the other preposition you use changes based on the word being used. So for instance, you would have "miedo a las serpientes" = fear of snakes, but you would have "amor por las serpientes" = love of snakes, and "miedo/amor de las serpientes" = the snakes' fear / love (i.e. the snakes are feeling these emotions.)
Russian does something similar, i.e. you have "страх перед змеями"="fear of snakes", here "перед" is a preposition that often translates as "in front of", which makes sense, i.e. you feel fear when you are in front of a snake, whereas you have "любовь к змеям" = "love of snakes". Russian has cases, and interestingly, neither of these constructions take the genitive case, but rather, the instrumental, a case that doesn't really correspond to any case in Latin or Greek, one that tends to be used for a means that brings about some sort of result. In Russian, however, you can also use the genitive case. My Russian is clumsy so maybe a native speaker could step in and explain this but I'm pretty sure that Russian does have some constructions with the genitive that are at least somewhat ambiguous similarly to how English can be with the word "of", and how ancient Greek and Latin can be with the genitive.
What this is getting at, is that this stuff is incredibly idiomatic, and not necessarily handled the same way in languages of the same language family.
There aren't really any techniques to distinguish between these two uses in the cases that they're grammatically ambiguous. In this case, you really need to look at context, and there are, unfortunately, cases where the context is ambiguous. And in these cases, these phrases can amount to bad communication! Remember when reading ancient texts like ones in Greek or Latin that those authors were human just like us, and sometimes made poor word choice leading to ambiguity, just the same way we do! Other times, though, especially in poetry or rhetorical language in speeches, they might deliberately use ambiguity as a literary device. I'd be curious if someone can come up with an example of genitive being used ambiguously as an intentional literary device...I bet such examples exist!
