What is the difference between an Acting President and an Actual President?

[PresidentConfer]

Both then Vice President Bush and former Vice President Cheney acted as President when President Reagan and President W. Bush, respectively were unable to perform presidential duties.

My question is, what is the qualification to get a number next to your name as President? If Obama is hospitalized, Biden would only become acting President. So how did people like Tyler or Johnson become President?

As a follow-up, if President Obama, President Trump, or any other future President dies during his (or her) last week as President, would the Vice President become an actual president or an acting President.

Hypothetically speaking, President Obama dies on, lets say, January 10, 2017. Would we see the 45th President of the United States Joseph Biden, or would we see 45th President of the United States Donald J. Trump?

I know that this is a lot, and I apologize, but I'm very interested in this topic.

[PresidentConfer]

Acting President = someone legally exercising full presidential powers despite not legally holding the office. Acting President can either serve as a stand-in (with Actual President incapacitated) or during vacancy (some argue that a Speaker replacing President would formally be "Acting President").

So does it all depend upon taking the Oath of Office? If so, is it hypothetically possible to see a week-long presidency of Biden, Pence, etc?

[PresidentConfer]

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It is hypothetically possible.

Even without taking the oath, the President-elect becomes President at noon on the 20th of January. That is why, despite claims to the contrary, David Rice Atchinson did not become president for a day because General Taylor refused to be sworn in on Sunday March 4, 1849, but waited until after the usual Christian sabbath.

What is President-Elect Trump happens to die before Jan. 20? Would Vice-President Elect Pence become President on Jan. 20?