Well-known information about sound changes would tend to focus on only those most obvious changes like Grimm's law where the consonants change in all contexts. But that's like saying that vanilla, chocolate and strawberry are the only flavors is ice cream in the world. There are far more changes that are contextual, even if you don't as often hear about them. Aside from fun facts that might stand out to non-linguists as interesting, the science of historical linguistics is
all about the contextual changes!
And actually, Grimm's Law is NOT an example of contextless change. The sounds did change across all environments, but if you look at the details sometimes there was contextual variation. The 'voiced aspirated' bh/dh/gh series (whatever the phonetic realization was) actually became EITHER voiced stops b/d/g or voiced fricatives β/ð/ɣ depending on context.
Completely contextless changes are very rare.
I've always assumed that contextual sound changes were more well-known only because they produce grammatical peculiarities.
No. Sound changes are studied in order to know how languages change, how they are related, and what a plausible reconstruction might be.
You can't really blame me, the only time I've seen formal notations of conditioned sound changes is when I watched the videos of a blogger called NativLang.
But you wrote "actual linguists"! I'm not blaming you for your lack of knowledge (or for anything really), but you should be careful not to state non-facts as facts. Clearly you would enjoy and learn a lot from taking a class about historical linguistics. There are things you don't know, and that's really not a problem, but the solution is to read a textbook or take a class, not state non-facts on forums.
Can you quickly explain me what notation is used in research papers? If it's easy to reprogram my game to include it, I'll do it.
It's not too different from what you wrote, but it has different conventions. Phonetic symbols (usually IPA, sometimes traditional symbols from historical linguistics research, similar to IPA but a little different such as š for 'sh') are used, and there is a specific notation for the contexts in which the sounds occur, even down to specific phonetic features (sub-segmental level). You should take an intro class about phonology (using the traditional 'SPE' [= Sound Pattern of English by Chomsky and Halle] method even though that is more or less rejected today by phonologists-- it's a useful notation and a good introduction to relevant topics), as well as one about historical lingusitics (which will usually focus on sound change). Or if you can't take a class, read a textbook. You can also of course look up examples in research papers or on Wikipedia, etc. There are some examples here:
https://en.wikipedia.org/wiki/Sound_change (there might be some inconsistencies or oversimplification there to make it readable to nonspecialists; I just skimmed)
Trying to 'finish' the game before you master the basics is putting the cart before the horse, as they say.