Actually, no. But whatever.
So where are you getting your definition of genius?
Actually, no. But whatever.
I'll happily admit to the second part....where do you get the first part? "The common usage of words is also supposed to be a focus of 5e". Was that one of the design goals? (Not that the answer is relevant; there were many design goals that people feel strongly were not delivered upon.)
You are conflating narration ("fluff") and mechanical change. It's as simple as that.
Speak for yourself!
In the typical D&D campaign, how much time actually passes "on stage", where checks are being made and actions declared to which a PC's stats will matter?
Your objection strikes me as similar to the objection against 4e encounter powers, that it makes no sense that a fighter can (say) disarm only a single enemy without a short rest.
Of course that would make no sense as a principle of causation, but that's not what it is. All the encounter power tells us is what happens in the N fights where we saw that fighter in-action, "on screen" as it were.
In those on-screen moments, the player (or the table) can always come up with a reason why the character's genius doesn't shine.
Well, actually, I'll readily admit that. I'm afraid that if I'm restricted to "commonsense" definitions (which is what Maxperson has doggedly insisted is the only possible truth) then that eliminates all sorts of interesting narrative possibilities.
I'm going to haul this thread back on topoic and ignore posts that argue for the sake of arguing that the question shouldn't have been asked.Obviously inspired by that other thread, the goal of this thread is to describe characters who are geniuses, but for whatever reason they often fail at tasks that, in game terms, require a dice roll with an Int modifier.
How might you roleplay it?
Here's a backstory for a genius with an Int of 5:
Merander of Threbes (Male Gnome Wizard-1/Rogue-17), Professor of Logic and founder of the School of Innovative Thought at Sageville University, is a well-known genius. Ask any of the sages who work for him, and they will tell you so and give many examples of world changing discoveries and inventions he has to his credit. He is best admired for breaking 2048-bit encryption with a carefully-balanced clockwork teaspoon.
He has an Int of 5. How does he do it? He gets other people to do his research and solve problems for him, then takes the credit, that's how. Not blatantly, of course. He doesn't even realize he's doing it, but that's how it works.
He has a maxim that he impresses on his students: any idea that can't be explained simply, is wrong. He sets his students difficult problems (DCs up in the low 30s) of the type where finding a solution is hard, but once a solution is found it is easy to see that it is correct. Creating the formula for a new magic item, for example, is hard but it's easy to test it by making one and seeing if it works.
When a student comes to him with a possible solution, he challenges the student to explain it to him in simple terms. He role-plays and pretends to be a genius pretending to have an Int of 5, which isn't difficult for him, and if the student can explain his solution well enough for even him to understand it, it gains his approval. His technique is to say "let me see if I've got this right" and paraphrase the idea, adjusting the paraphrase until the student agrees that it is a correct paraphrase.
He then sends the student away with a new problem while he, the professor, starts to publicize the paraphrase (which is genuinely his own paraphrase) whilst omitting to mention that it wasn't actually his idea that he was paraphrasing.
He's been getting away with it for years.
An Intelligence check comes into play when you need to draw on logic, education, memory, or deductive reasoning.
Merriam-Webster.
Simple Definition of genius
1 : a very smart or talented person : a person who has a level of talent or intelligence that is very rare or remarkable
2 : a person who is very good at doing something
3 : great natural ability : remarkable talent or intelligence
...how can you reason well, without being particularly good at logic?