The brouhaha about if the POTUS-Elect is accepted as "my" or "your" president not only is an example of immaturity and unsportsmanlike conduct on the part of people who we should all expect better of, it is a case of missing the point entirely.
The U.S. Constitution created offices in which office holders are elected by various means to carry out the duties of said office. The authorities and "powers" to carry out said duties are inherent to those offices. Each person we elect to offices wield the powers of the office while in office but do not retain those abilities after no longer being in office.
What this illustrates is that any representative we elect is not "our", "my" or "your" president, congressperson, senator, etc.... as much as they are "the" officeholder. They personally do not have any more or less power or authority than any other person. Everything they are able to do is a power of the office.
One thing to keep in mind is that being only human, it is so easy to make easy assumptions especially the longer one is in such a position to wield such power. Such assumption often leads to an inflated sense of entitlement. People who spend too much time in office begin to think of themselves and associate with others who look upon them as the embodiment of the authority and powers of the office. They use terminology that assumes personal possessive ownership like "my" and "your".
It is because of such conflation of concepts such as being an officeholder and a personal ownership of power that makes the concept of term limits attractive to the electorate. Wielding power for extended periods of time, especially when little to no "real" challenge is posed not only corrupts but is intoxicating and addictive as well.
Few people are seen as strong enough to let go of power when it is time. The character of someone to willingly relinquish it is often considered ideological but rarely achievable. Thus, because though we say ideologically that term limits, thus limitation of individuals usurping powers of office, lies with the electorate making responsible choices, constitutional limits on terms is often thought to be the more realistic and practical approach to reducing or trying to eliminate corruption and usurping of the powers of office.
Personally, I am idealistic enough to think that there shouldn't be a need to impose such restrictions such as constitutional term limits on officeholders. I am also pragmatic enough to accept that far too many people lack the strength of character to hold themselves and others to such ideals.