Hmmm... I have to say that now you are pointing very interesting theories... Olivia's surprised internal thoughts always annoyed me a lot. I didn't believe for a second that she's acting even to herself. That made no sense at all...Whenever reviewing, considering, or analyzing a theory, one must always look at the evidence, and see if they support one or all of the plausible theories. If one theory exclusively holds true against all evidence and challenges, then it is definitively true, but if a theory's evidence is shared with its contrary, or there remain contradictory issues with the theory, then it must remain a theory, alongside all other plausible theories. Therefore, I have gathered all the scenes pertinent to the Agent Olivia and Intelligent Olivia theories.
Agent Olivia = The Olivia that knows EVERYTHING before Alexander's death, and is feigning shock and ignorance.
Intelligent Olivia = The Olivia whom Callista has raised and trained for the upcoming Gambit, but from whom Callista has hidden most of the truth, likely telling Olivia that KG wants "the best and brightest" in the world on their executive staff, and since Olivia matches that, they will come for her, and she has to either excel, or "die". This Olivia applies her super-intellect, analysis, and deductive reasoning to use any gained knowledge to discover strategies, plots, motivations, and secrets.
To save space (and time) rather than screenshots, I have copied the lines from the crucial scenes.
Day 4: The Pier
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By Day 6, Agent Olivia has known about her father for years, while Intelligent Olivia has known for 2 days. They are also both adament about making sure that only Elena [and Cassandra] know that Olivia knows she's a Karlsson, either because of her own agenda (Agent), or under Elena's instructions (Intelligent). At the zoo, per your screenshot, Olivia apparently believes that the security system is programmed to her true last name: Karlsson. She muses (in the screenshot) whether Veronica invokes the "first names only" rule to hide "Olivia Karlsson" just from Kane, or from Olivia as well, and thus wonders if Veronica still doesn't know that Olivia knows her true last name. Neither Agent Olivia nor Intelligent Olivia wish to be discovered to already know that she is a Karlsson, so both would have this thought process.
Helicopter Flashback
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Hike Flashback:
Kane states that he asked Callista about "Dad", but she denied him. He does not infer/suggest Olivia try, but Agent Olivia, who knows that "Dad" is not her father, needlessly and unprompted says that if Callista won't tell Master, then there is no hope for her or Seth to ask about "Dad". She also asks Kane why he thinks Callista has been so secretive of her past, when Agent Olivia knows why, and also knows that Kane's ignorance is [apparently] paramount to the plan. Olivia's other comments in this scene also support both an Agent Olivia who has had recent training with Alexander, and an Intelligent Olivia who has just been informed that her future is with KG, in which she will have to perform in order to survive, or "die"... a very fatalist destiny that would turn anyone somber.
End of Day 6: The Mother
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I made the assumption that Tess used the "unreliable narrator" card to save the surprise until we are revealed the truth at the helicopter scene.
But now you've made me doubt about everything I thought about her
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