Absolutely! Dogs use yawning a lot in their communication if you look close enough. They also have other messages like smelling around, looking away, etc. I think they all might mean something along the lines of "I don't see you as a threat to me so we are good"
They're called calming signals, and the reason they're calming is because they make it obvious that the individual isn't "on alert".
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".
Cats also regard looking away as a friendly gesture. Which is probably one reason they anecdotally tend to seek out the company of the one person who is allergic or does not like cats. Humans not accustomed to cat body language tend to think of looking away as a sign of aloofness and disinterest instead.
Some people jokingly say that cats are autism and dogs are ADHD and if you know people who are autistic or have ADHD, the parallels are fairly obvious. Autistic people also struggle with other humans insisting on eye contact.
As do Finns (I'm one), and people of many other non-American cultures. And many non-autistic but shy or introverted or highly sensitive persons. You seek eye contact momentarily when you make a point in a conversation, or respond to a point made, or want to empathize, or whatever. Prolonged or too frequent eye contact is weird.
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.
