Lame-duck as I’ve seen/heard it used (40+ years in the DC area) means a President who is nearing the end of a second term, and therefore can not run again. There is no personal insult intended, it’s just describes the last year or so of the eight-year stint.
Given the manner in which things are accomplished in DC, a lame-duck has nothing to trade, and can not be depended upon to provide any future benefit. There is also strong pressure from his/her own party to avoid major controversy which might affect their ability to elect a party-member in the next go-round.
It is also true that foreign countries are aware of the short shelf-life of any promises made by a lame-duck and will therefore prefer to wait for the next President before finalizing any major negotiations.
Some Presidents may view this time as one of unprecedented freedom, a time to push through highly controversial projects, and many a pardon is requested during this time period. They are still lame-ducks, regardless of how active they may be during that time.
So, given the definition as I’m accustomed to hear it, anyone who is elected to a second term becomes a lame duck. Your question struck me as the equivalent of a query on the President’s health.