
On a related note, I'd like to highly recommend Lucy Suchman's work, mentioned in this article as Plans and Situated Actions: The Problem of Human-Machine Communication, but the updated version now called Human-Machine Reconfigurations: Plans and Situated Actions. The new version has several extra chapters and some other revisions. I've read it several times, and had my mind blown each time.
https://www.goodreads.com/book/show/342380.Plans_and_Situate...
If we're actually having some kind of debate about how to signifiy in an interface I'd probably choose "profile" and forego "my" and "your" as they imply a relationship that does not exist and a facility in the interface that is not possible.
Given that most people aren't actually concerned about that and just want their stuff to be "nice" or "friendly" I'd pick one of "your" or "my", whichever fit in the grammar of the rest of the system, and be consistent.