I gotta say, I don't get this.
Part of what informs the immersive roleplaying you're embarking on, be it a character on a stage or a character in an RPG, are the stated or pre-set attributes of the character you're about to play.
In a stage-acting situation the script notes often have some guidelines as to what (in theory) makes a character tick, and if they don't the director sure as hell will.

Maybe it's noted in the script that your character walks with a pronounced limp, for example; or that he's 95 years old and hard of hearing. You-as-actor are then more or less expected to use these guidelines - perhaps among other things - to inform how you approach portraying the character.
Same thing goes in an RPG, only here those informing guidelines don't come from a script or a director but instead are provided by some numbers on a page.
I think the two media are very different in this respect, at least potentially. In stage/screen acting, unless we're talking about a total improv, the decisions about what happens, how the story goes, and what actions the character will take within that story have all already been made before the actor plays the role of the character. The decisions left for the actor to make (perhaps, as you say, in collaboration with a writer and/or director) are those regarding characterization and portrayal, mechanical things like how the limp will be executed or how the character's age and deafness will be communicated to the audience, and most importantly, at least to me anyways in the way I approach acting, decisions about the character's inner life, answering questions about what motivates the character to take the actions that the writer has decided for it to take. By becoming immersed in the character's inner life, the actor can achieve what is referred to as being "in-character".
Improv, with which I have very little experience or aptitude, works a bit differently and might be comparable to playing an RPG where the resolution mechanic is nothing more than "Yes, and..."
To me (and this is where the
potentially in my first sentence above comes in), the heart of roleplaying in a roleplaying game is not characterization and portrayal. It is deciding what actions my character takes in response to situations in the game. In this way, a player in an RPG is more analogous to the writer of a play. The resolution mechanic alternates whose authorial decisions prevail over the resulting fiction, replacing the sole-author of a play with the game's participants. I don't see any clear distinction between roleplaying my character and authoring my character, so just as an author can be said to write a character from an in-character perspective, feeling-out and exploring the motivations and inner-workings of the character as the story develops, I find it most immersive to roleplay my character, as in
decide my character's actions, from that perspective in the moment of play.
An actor in a play is not deciding what actions the character takes in the moment. This is what learning your lines is all about. You memorize the text of the play so it becomes internalized and can be motivated and actualized during the performance. This is not roleplaying in the sense of playing an RPG.
A roleplaying game, for me, is not a performance. It can be, but that isn't the primary activity of roleplaying. The primary activity of roleplaying is an exercise in authorship in the form of a game. I, the player, get to make up the actions my character takes. Treating the numbers on my PC's character sheet as a sort of pre-authored script to be performed that limits the actions my character can take would be abdicating my responsibility as a player to play my character and having my character just play itself. For me, that wouldn't be immersive roleplaying because then I'm not inhabiting my character. No one is.