Earlier quoted context omitted.
I'm not sure what point you're making, but it sounds like this is less about how you define the self and more about how you define involuntary. The heartbeat is an involuntary action and is simultaneously part of yourself. That's all I mean.
I agree that a heartbeat is an involuntary reflex but I don't see how you can compare that to the mind chatter which is AFAIK related to cognition much like the rest of your thoughts and mental processes. Not only that, your inner monologue is completely unique to you. Probably from a part that you do not identify with which is why I was asking.
If you consider a young, attractive person who identifies with their beauty, eventually when they age and likely lose that beauty they once perceived, they end up feeling like they have lost themselves--or at least some huge aspect of themselves--and suffer greatly because of it.
Alternatively, if they recognize their looks are just a transient thing that is not their identity, then losing that perceived beauty is much more benign and life is more peaceful. It's because there was always much more to them than their perceived beauty.
It's the same thing with emotions and cognition. Identifying with mind chatter--or thoughts in general, in their supremely transient nature--is to impose a limitation on ourselves and invite suffering.
Not to pretend that I'm this untouchable entity that is above all of this, but I've found this to be a very healthy practice that makes space for a peaceful, harmonious way of living.