I have the impression that there are multiple related things called "immersion." Some examples I see, like this, make sense if "immersion" consists of a strong concentration on the character's background and thinking, trying to frame everything in terms of their personal psychology and drama.
That's not what immersion means to me. Trying to relate everything to background and psychology feels more like soap-opera to me. What I try to do with "immersion" is inhabit the entire person of the character. That includes background and psychology, of course, but includes their skills and abilities, and all their other available resources, be they material, social or organisational.
So my influence over the fiction outside the character comes from the character doing things: making plans, buying or otherwise acquiring equipment, asking NPCs and the other PCs to do things and creating details that the GM has not specified, but seem plausible. This approach can be carried out without leaving "actor stance," or otherwise breaking immersion. It does require a fair amount of concentration, but it pays off, for me, in being more enjoyable.