Earlier quoted context omitted.
That's called meditation these days, but it's not about thinking no thoughts. It's about sorting your thoughts until there are no more worries. Everyone should be doing this.
Not quite, it's so much different from meditation. What you are referring to is more of the western concept of "quietism" which is pretty synonymous with meditation. But in either case, I agree, everyone should be practicing this.
It certainly sounds like one of you is talking about the meditating by using your thoughts as an object (sorting thoughts obviously makes no sense once you stop identifying them as "yours"), while the other is talking about simply doing it object-free.
Both are taught by many of the same traditions, with neither being necessarily better or worse, though usually the object-free version is taught later because it's hard to do without something to focus on when you're first starting.
I personally find the prior more useful when I'm trying to process a specific problem or stressor and the latter to be more useful after I've done some preprocessing. They are often done sequentially for this reason, as well as because meditating briefly with a specific focus helps prepare you mentally for the more general focus-free meditation where you simply rest in the experience of existence. But other preparation meditations with focuses can also be used to help you mentally prepare, such as focusing on body sensations, breath, vision, sounds, etc. Thoughts are just another thing you can observe.
EDIT: I'd note that the Wikipedia article on Hesychasm describes it as asceticism and "blocking off", so in retrospect I'd describe it as a different class of meditation than those that involve resting in experience. More like visualization practice or some of the scarier death practices. But they're talking about the "advanced" version there, and such things are easy to misconstrue because the language for describing it is poor. But I'd still call it a meditation.