Q:

What is the nature of a quorum?

As I understand it, a quorum is the minimum necessary for business to get done in a meeting.  My understanding is that a quorum merely prevents business from taking place without meeting the required number of members present.  It is not an grant of authority for things to take place when you do happen to meet the minimum threshold.  
For example, the quorum may be set at a majority of members.  But without a rule stipulating that a majority vote wins, a quorum is not sufficient to tell you whether or not a majority can carry a motion.  To go further, the quorum certainly does not override any requirement that a supermajority carry any motion.
In sum, my understanding is that a quorum shouldn't be read to explain what number is necessary to carry a motion.  My understanding is that a quorum merely denotes the number required for any business to take place at all.

A:

Quorum defines the minimum number of members present in order to conduct business. Once quorum is met, members may make motions and the assembly may conduct business.
Now your question concerns the minimum required for passing a motion, and you are correct some motions require more than majority, such as 2/3 vote.
The problem in your analysis is that rules of procedure assume that the majority or supermajority is met from the members present and voting, not from the total number of members.
Let me show you some examples so you can fully understand.
Let's say there's an assembly with 21 members, with the standard quorum requirements of majority of the members.
The minimum number of people required for quorum show up, which is 11 (more than half is majority).
In order to pass a motion, you only need majority of the members present and voting. That typically means 6 people. However, let's say 2 members abstain from the vote or are present at the meeting but don't record their vote. Then a vote can pass 5-4. Quorum is met and a majority of the members present and voting, voted for the the motion.
Similarly 2/3 votes are counted from members present and voting.
There is an exception to this. If the organizations bylaws say that it takes a majority of the members to pass a certain motion, then the motion must pass by the majority of the total number of members in the organization, which is 11 in this case. Thus all 11 people present in the room would have to vote for it.

