However given how many tools there are and how fast each tool moves, I find myself jumping between them quite often, just to see which ones I like most or if some tool have improved since I lasted checked it. In this case LNAI is very helpful.
Most prompts I do I execute in all four at the same time, and literally compare the git diff from their work, so I understand totally :) But even for comparison, I think using the same identical config for all, you're not actually seeing and understanding the difference because again, they need different system prompts. By using the same when you compare, you're not accurately seeing the best of each model.
Personally I put stuff like that in the readme, since it's useful stuff for humans too, not directions just for machines, and I'm mostly building for other humans. The lighter and smaller the AGENTS.md end up, the better the models are at following it too, from what I can tell.
Then for the application specific documentation, I'd understand you'd want to share it, as it stays the same for all agents touching the same codebase. But easily solved by putting it in DESIGN.md or whatever and appending "Remember to check against DESIGN.md before changing the architecture" or similar.