If the individual means you harm, they're going to be paying close attention to your every movement since they're about to attack. They'll be staring directly at you, sizing you up, preparing to pounce.
Looking away, smelling around, yawning, slow blinking or eye closing, are all boondoggles that signal, as you said, "hey, I feel at ease with you enough to do this idle busywork, almost as though you weren't here at all because I don't see you as a threat or target".
That said, cats very much do stare at your face when they want or are expecting something, usually food. They also have the adoring eyes-half-open stare mode you can see examples of at r/Catsmiring.
I would say that cats better fit the introvert archetype, and dogs the extrovert one, rather than something from the DSM. Anecdotally there's also a significant positive correlation with their respective owners' personalities.
But not always. For instance, a common sign of flirting in humans is to repeatedly catch someone's eye and then look away.