I agree, it's more akin to saying a King is elected by nobles...
Power sharing agreements between the powerful who are not accountable to the people isn't something I'd classify as "democratic".
I'd consider term limits to be anti-democratic anyway. A functioning democracy has the ability to boot politicians out, and the people within the democracy, not the law should be trusted to make that decision.
As how the US president is selected by the electoral college. The system used today is a far cry from the original scheme. Look closely and the original doesn't look very democratic, even by the standards of the time. Democracy is an idea, a collection of principals. How it is actually implemented varies considerably.
I was thinking about various Kings during the War of the Roses when I wrote that.
But a better comparison would be the "election" of a Pope. It's only among Cardinals, who tend to vote for a Cardinal to become Pope. Even though a vote is taking place, we are not talking about a democracy.
As to which is closest to an ideal democracy in practice: America, Vatican City, or China, the contest in my mind is who is in second and third place, not first.
The Electoral College is ultimately rooted in the American people. The number of electors is a direct copy of the number of Representatives and Senators in Congress, and the electors individually are determined by each state's government who are in turn elected directly by the people in those states.
The Electoral College itself exists as a way to help keep the Executive Branch separated from the Legislative Branch. Congress (aka the Legislative Branch) can step in in place of the Electoral College if the electors deadlock to a point of no recovery, but this is considered highly unusual and an undemocratic act because it violates the Separation of Powers safeguard.
The National Congress of the Chinese Communist Party has a history of just over 100 years and resembles other modern democratic structures like the Democratic/Republican National Conventions in the US. Voting is used to various degrees by party members and committee members. Many of the recent anti-democratic trends have limited who votes, and what information voters have access to. It is a shame that China is trending less democratic. It appears that one party systems is more susceptible to consolidation of power and cult of personality than western political systems.
There are varying degrees of democratic systems possible within the context of a government. In the United States for example, Federal Senators used to be appointed by State legislators instead of being directly voted on by the people - in this case, a political body (State legislators) operated within a democratic framework to appoint Federal leaders.
There is a body of representation in China which votes on the next leader, which has rules to limit power within that system.
In the usage of the term "democracy" meaning "a group of people voting on outcomes" - China has democratic traditions within it's political system involving voting and consensus and limits of power. Part of these traditions included term limits on leadership which has been removed by the current leader (again, not unlike the way Putin removed term limits in the precursor to military adventurism).
