I thought I was using "the subject" in the same sense as you used it when I was last at your house. There's no sense of agency with the subject, so it's not a sense of self as doer. I'm not sure that it's different from the "watcher." But I didn't think we were talking about the "watcher" when we sat in your house.
Can you have a subjective sense of the "watcher"? Can you apperceive it ?
Yes you can. But it is kind of mysterious, much like getting a meeting with the prince in Kafka's Castle.
You see the footprints of it everywhere. You feel its presence on the other side of phenomenality.
Then one day something happens and "fall back" into the mystery, and suddenly the world reappears in a new light as you realize you have become the Self--at least as an instantiation of sentience within the limitations of flesh, and you always were the self, but did not know it because you expected something different and perhaps more transcendental.
Then one day too, you see the Self of All in its infinite glory, and see the oneness of both.
The area you are in is very subtle and can be infinitely confusing trying to separate the concepts about what you perceive and the perceptions themselves. They color each other in complex ways that you have to tease out your own self. Later, you will discover that both the direct perception of Self, and your description and concepts about it will change as you grow until you stop talking about it and just rest there as the mystery of you unfolds.