My character has a habit (or running gag) of pointing out plot holes in conversations with her friends who lie a lot. She is quick to tell them what fragment of their stories about what they had just done recently doesn't make sense. She knows when they are lying about something as if she is Sherlock Holmes.
But Sherlock Holmes was made of two things according to a Wikipedia article of the character's category sections
Which is my character most likely to be clarified as?
Sherlock Holmes is famous for deducing answers to puzzles from observation. He was widely and deeply read, although he also deliberately forgot information which he felt wasn't important. He was a detective because the matters brought to him were problems which needed investigating — often crimes, but not always.
An intellectual is someone who is intelligent and studied, but not necessarily someone who is greatly observant or someone who can deduce facts given those observations.
So if your character knows that Bill is lactose intolerant but he says he was drinking milkshakes with Suzy, she can call him out on that lie. That's being a detective. If Bill starts going on about the benefits of supply-side economics and she counters with examples of Kansas's distastrous budget, that's being intellectual.
