If a US vice president is installed as president due to the death of the current president, does that count as the vice president's first term?
The issue is if some one can actually serve as President of the United States for over 8 years. If a vice president is seated as President during the 2nd year of his term as vice president, can he be elected to the office of President in the next two elections, potentially giving him over 10 years as president?
Yes, a person can serve for more than 8 years. This is spelled out in the 22nd Amendment to the Constitution.
What that amendment specifies is that a person may serve 2 years of someone else's term and then be elected twice on his or her own. That would mean that the person could serve for 10 years total. If the vice president serves more than 2 years of the previous president's term, he or she could only be elected once.
So, a unfinished term only "counts" against the vice president who finishes it if there is more than 2 years left on that term when the vice president becomes president. This is why President Lyndon Johnson could have run for president again in 1968 after having served a little over 1 year of JFK's term and then one of his own.