For me, I draw the boundary based on latency, unconscious guidance, and predictability. I can send a thought to move my hand, and it moves as I think of it. The hand is within the boundary of "me". Holding a pencil, I do not need to consciously consider how to form each curve of a letter. The pencil is within the boundary of "me". I can predict what emacs will do when given keystrokes, so emacs is within the boundary of "me".
On the other hand, there is a large delay between deciding to open a door and it responding to my pull, so it clearly is not "me". I need to consciously consider what search terms to query, and cannot do so at an unconscious level. Even when I repeat the exact same query as I did a year ago, I may not find the results I was searching for, and so there is no predictability. These lead me to feel that a search engine is not within the boundary of "me".
Your door example raises further questions. When I pick up a remote control, the control moves just as easily as my hand. Does the remote become an equal part of me immediately?
It also means that it depends on the situation. If I'm reading a book, then the movement of my hands feels instant as I reach to turn the page. But back when I played the piano, at times my hands could feel like they are falling behind, not listening to what I'm telling them to do, responding too slowly for what is being demanded of them. Whether or not my hands feel like "me" depends on what I'm trying to have them do.
> Does the remote become an equal part of me immediately?
I've never really thought of it in terms of time, in part because it is only something to quantify in retrospect. A pencil in my hand feels like a part of me, that responds as I move it. If I press it against a piece of paper, I interpret the sensory input as "felt" from the tip of the pencil, even though I know that I have no nerve endings there. But a pencil on the table isn't part of me. Whether there's a smooth transition between the two as I pick it up and gain control over it, or whether it's something that just "clicks", I'm not sure.
Meta's avatars (and many other ingame models representing a human) would have you believe so, since they don't have legs.
If one is to read Wikipedia and repeat it verbatim without understanding then one is only a single component of a greater transportation medium. But if in contrast, if one was to internalise those words and draw lines of inference between ideas so elusively captured therein and a wider base of knowledge then maybe you are something more.
But perhaps he does not care about the far future as much.