I've not done particularly rigorous testing, but I've done this a lot with Claude to get a feel. What I've noticed is for certain open-ended tasks, Claude is extremely primeable: it will pick up on minor differences in wording in your prompt and run with them hard.

It can be frustrating. The AI pretends to be a human, and so a part of my brain expects them to commit and have a "parti pris" like a human, so the exercise is a good reminder of the feedback loop. My mental model is that before the first three or four messages, the model has many finer points of its personality still underdetermined. I'd suggest that as the mechanism for "role-based prompting". And it explains the "savant sleeper agent" thing you describe. You want to get the state in the right attractor on the manifold.

These machines are pretty incredible, but for conversation-driven workflows you really have to be in the driver's seat. A human has a property that the AI does not have, at least under current architectures: we are regulated by the outside world. A bit of a tangent, but I can see how AI psychosis arises from these dynamics.