Moraine’s life's mission is to find the chosen one, known as the Dragon Reborn, and deliver him or her to a final battle with the Dark One. But this mission wasn’t quite so simple.
We learn later in the prequel novel New Spring that Moraine has taken this mission after refusing the direction of the Aes Sedai Amyrlin, Sierin, to remain, essentially forever in duty at Tar Valon — a fate Moraine considered to be worse than death. Moiraine literally has to run away to escape being imprisoned by her usefulness to the Tower. This has brought her the ire of the leadership.
In this regard, Moraine isn’t just keeping her Aes Sedai status on the downlow from just the Whitecloaks. She also has interest to keep her own identity a secret from the prying eyes of the world as well, because it may have consequences should her whereabouts make it back to the
White Tower while she does not yet have a Dragon Reborn in hand to excuse her behavior.
However, within the Wheel of Time novels, at this point in Eye of the World, we hadn’t yet been treated to the events of New Spring, and the issue primarily driving Moraine’s behavior is the fact that around Baerlon, Moiraine and Lan are known under pseudonyms, and folks are on edge because of the presence of Whitecloaks; this was the first foreshadowing of the dangerous relationship the Whitecloaks have with the Aes Sedai.
Additionally as @Amarth noted, the most obvious party Moraine tries to avoid detection from is Dark Friends though, since there are trollocs out searching for them.
This interaction is not anything that is precisely adapted from the novels. The first meaningful novel interaction between Moraine and Whitecloaks is when the group attempts to leave Baerlon, and in vivid manifestation with
Morraine appearing to growing tall to threaten the Whitecloaks — growing so tall she can simply step over the walls.
so as such, this scene in the show may be intended as a subtler composite manifestation of magic than what exactly occurs in the novels.