@jwayne I would argue that consciousness, in general, has a tendency to identify with what you defined as the 'second-level' of thought. The realm of mental formations is where consciousness tends to lie in at a given moment (getting caught up in thoughts, daydreams, memories, future plans, etc.). The 'first-level' of thought is mostly managed by the subconscious (or even unconscious). Only by bringing the direct consciousness experience to this first-level of thought can the individual become aware of the false self and start detaching from the false experiences of the self.

The first-level of thought is typically the direct present experience, which is then stained via the initial processing that occurs in the mind. And this initial processing occurs in a fraction of a second (sometimes as fast as 0.1 seconds, sometimes longer). This initial processing is how we color and conceptualize the world (i.e. how we form our impressions and responses). An example of this is when we see delicious food, we label it in a positive light and react in a way that gets us hungry. Or when we hear a startling noise, we label it in a negative way and our body activates the fight or flight response. So, this initial response is what dictates our subsequent human behavior (based on past conditioning) and is responsible for a lot of the manifest content that the mind experiences.

Mental formations (memories, ideas, streams of thoughts) typically stem from the prompting of subtle subconscious reactions. Which is why that unless you sharpen the mind to become aware of these subconscious processes, the true self cannot be accessed. And the true self (i.e. state of desirelessness) is what comes to light after the individual becomes aware of the mind's initial processing schemas and transcends them entirely.

So on that note, Desire would be a state where thoughts are seeking their own propagation, not their termination (i.e. Samsara). The unenlightened individual cannot conceive of the self without external support. This is why the unenlightened individual relies on the stream of incessant thoughts, and the object of said thoughts (external objects in the world) as a proxy for their lack of self. The state of Desirelessness would be a state free from the gravitational pull of thoughts (similar to the Zen concept of Mushin, or No Mind). Only in this state is the individual aware of, and sustained by the true self.