Leader of the world and leader of the free world are two different things. As far as I know, no country in the past century has called itself either of them. I would not be surprised if the title was applied to countries such as the Imperium Romanum and 中國 in the distant past. Even more distant would be possibly Greece given that it’s leader was called the King of Kings around the time of Troy.

However, I believe both titles are applied to specific office holders in the modern day. The U.S. president is called the leader of the free world by many. I believe the title leader of the world is bestowed upon the Pope during Papal inaugurations.

With all due respect, you're being pedantic. No, we're not talking about official titles bestowed from on high. The US has been considered the world leader from WW2 until sometime in the 21st century, partly because nobody else wanted to claim it, partly because American foreign policy is aggressive, partly because the American economy was the most robust on in the aftermath of WW2.

The U.S. has never been the world leader. The U.S. has been a military leader for other democracies since the end of WWII. That is why the title, leader of the free world has been applied to the U.S. president, who is the leader of the various military alliances between various major democracies. If you claim the U.S. considered itself to be the world leader, then you are claiming the U.S. considered itself to be the leader of the Soviet Union, the Warsaw pact, North Korea, etcetera. That is absurd. The U.S. likely did consider itself the leader of Vietnam and Cuba before the democracies in those countries fell, although that was in a military capacity. That is why the U.S. intervened militarily in both, although both interventions failed.

Saying that the U.S. got its leadership position because no one else wanted it is historically incorrect. Following WWII, every other major democratic power was in ruins while the U.S. was the sole major power left in tact. Without any military attacks on U.S. soil, U.S. military strength had skyrocketed during the war. As the war progressed, the U.S. attained the status of a great power and by the war’s end had become a nascent super power. U.S. strength continued to grow after the war due to the threat of the Soviet Union. Meanwhile, the other great powers never fully recovered militarily since they focused on their economies while relying on the U.S. for security. The U.S. gained its leadership position because no one else existed that could claim it.

If I say, “I like to lead by example”, that can mean leadership in the moral sense of progressive politics. Americans view America that way, as the leader of liberalism and social justice. Even Americans who don’t see America or American history that way, tend to see themselves (ironically?) that way. But that’s a different meaning of leader. We lead by things like protests, humanitarian aid and having enlightened celebrities.

Provide a source on the Pope thing please.

The USA has, by any measure, a veritable monopoly of coercive force on the world stage. Their military expenditure is more than the next 15 countries combined.

This means they have the capability to enforce their will globally to a significant extent. In an arena such as geopolitics, justice is the will of the stronger, no holds barred. This makes the USA arguably the primary concern in geopolitics, the ring you need to kiss to do anything on that stage.

Keep in mind that “claimed” may be referring to the sense of “won” rather than “stated”.

From that perspective it’s not too much of a stretch to call them the world leader, but that does ignore the fact that leadership implies the will to lead and to a significant extent the requisite wisdom and skills.