I was taught at an early age in the USA that when we write about our President, we are supposed to capitalize the title in order to signify that it's on the federal level. Is it correct to always do this? And what about saying "the presidency" -- should that be "the Presidency"?
I learned that in school too! But I don’t think many people actually follow that rule (which means it isn’t much of a rule).
AP style does not have this rule. It prescribes what mgb suggests in an answer below: capitalize only when the official title precedes the name. President Clinton has served as president since 1993. This style seems most common in practice.
The Chicago Manual of Style agrees, as Sven Yargs points out in the comments below.
Some writers, maybe 5% to 10%, follow the rule you described.
The question of which is correct ultimately hangs on what you mean by correct. This is one of the questions where there is no strong consensus. There is no ultimate authority, no Supreme Court of Capitalization, to settle the issue.
However there is a consensus on one point: in sentences like Thank you for joining us, Mr. President, it seems President is always capitalized.
I've been continually frustrated by the general consensus against capitalizing "president" when discussing the POTUS. Perhaps it is because as a law student, I read hundreds of Supreme Court opinions, which (as far as I can recall) always refer to the office as "the President." This applies whether the reference is to a specific act, such as "the bill was vetoed by the President," or a general description, such as "the sole authority rests with the President." This capitalization seems useful and appropriate to me because the President is a legal office that carries legal authority in and of itself, to say nothing of the fact that in our system the title is itself an entire branch of government.
To say "president Barack Obama" is to imply he has been picked as some sort of leader of a discrete group. To say "the President" is to acknowledge the lawful power granted to him who holds that title solely on the basis that he holds that title.
The US Constitution (the document which actually defines what is a US President) uses "President" when referring to POTUS and "president of the Senate" when referring to the list of responsibilities of the Vice President. Since it is the founding legal document, all legal document can be expected to use capitalization when refering to a title and to not use it when using it as a job description.
As for the specific title of the US President, I found the answer while researching my own question on this subject here. Apparently NY Times switched from capitalizing to non-capitalizing spelling around the time of transition from Clinton to Bush (in 2000). I put the links to quotes in the body of my question there (in case anyone is interested).