We also earned that Jaresh-Inyo was "elected" for President, but we don't know by whom.
President: " I was content to simply represent my people on the Federation Council. When they asked me to submit my name for election, I almost said no."
"Leyton: "Hardly a dictatorship, Ben."
Sisko: "Overthrowing a legitimately elected President...?"
So... We know the President is elected, and we know a Member of the UFP Council represents his people there. But we don't know if a Council Member gets elected, and we don't know if people are involved in the election of a President.
I'm not aware of any other issues or positions in the UFP being open to popular vote. However, voting was part of the process where Ambassadors (and possible other officials and reps) decided whether to accept Coridan's petition for UFP membership in "Journey to Babel". The vote was to be cast in a special emergency conference, which probably isn't the same thing as the UFP Council, and it appeared every member planet got one vote. This might be considered representative democracy at work, if only we had some evidence that the Ambassadors casting the planetary votes represented their people via a democratic election process. Just as with Council Members, though, evidence for such is lacking. And if anything, "Ambassador" sounds like somebody who is appointed rather than elected.
None of this is solid evidence for the lack of democratic mechanisms in the UFP, though. It is quite possible that democracy is being employed in legislation (probably in the representative form, given the existence of the Council and its Members) and in execution (since we know virtually nothing of the executive branch of the UFP, unless this in fact is the Council), and perhaps even in the judiciary process (where judges might be elected rather than appointed, and/or the process of judging actually conducted by popular vote).
It would be fun to speculate on the subjecting of legislative, judiciary executive powers to direct popular vote - certainly an option for a society with advanced means of communication and processing of information. But since Star Trek drama calls for officials of various sorts, it appears likelier that Trek democracy, if any indeed exists, is of the representative sort, and the President is an actual executive official with powers surrendered to him by the people.
Well we know very little about how the Federation Council works. Only what can be inferred from the vague referrences in episodes.
We don't know for definate even how the United Earth Government works, from what can be inferred from ENT is that it appears to be a Parliamentary based, but is that with an elected head of State (President) with a Prime Minister being the Head of Government.
1.>For all we know the Federation President isn't elected by popular vote but by the council members themselves.
2.>Each member planet maybe able to choose how it's selects it's council member, Whether that be by popular vote or by vote of the elected government of each world.
3.>It is possible in the 22nd-23rd century era, that the Ambassadors fufilled the role of Council member. Or the ones that we saw in TOS such as Ambassador Sarek did double duty both as Council Member for the Vulcan People as well as Ambassador to Earth. Moving into the 24th century era with the larger Federation some changes were made to how the Council operates.
It is just as likely that the position as Federation President doesn't come with any special privelages i.e a power of veto. So any vote comes down to the simple 50%+1 rule.