That's not quite what I meant. I said that the party would feel it, not merely notice it. I meant with this that the consequences of the Tank messing up in this manner are not only obvious but severe as well.
You expressed a concern about vagueness. I certainly didn't mean to express such sentiments. Let's see if I can help out with that.
Enabling the rest of the party to perform their roles efficiently: This may be better if defined as a "do not": Do not do something that prevents a party member from performing their primary duties. Positioning is arguably the biggest aspect of this from a Tanking perspective. If the monster is constantly moving, no DPS will be able to efficiently DPS. If the monsters aren't positioned quickly, you're delaying the kills. If you spin a monster in circles, the melee DPS cannot hit their positionals. If you run out of the range of the Healer's spells, they can't heal you. Tanks quite simply have the most methods available to them to directly interfere with the party's ability to perform their roles.
Ensure that the fight is as easy as possible: Easy is, in most cases, certainly subjective. We'll need to focus on more objective examples of this then. The obvious example in the context of this thread is that being hit in the face without a Tank Stance on makes you more difficult to heal. Whether the Healer can handle it or not is irrelevant; you would be objectively easier to keep alive if you were in a Tank Stance. Another example is cleaves; don't point them at the party. Unnecessary damage increases the burden on Healers. Other more situational examples would be taking over mechanics that don't explicitly require a Tank to perform them but would, for example, give DPS more uptime if you took care of it instead.
Now, you presented one situation that makes things more interesting: slipping out of the Tank Stance in order to beat a Hard Enrage. Enabling the performance of roles has no compromise, but this is an example of a time when keeping the fight easy can be bent. You misunderstand when you posit that the ability to win is equivalent to making the fight easier. Situations like those barely there enrage wins are the result of one of the most important aspects of undergeared progression: calculated increases of risk to the party(making the actual performing of the fight more difficult) in exchange for gaining the DPS necessary to win. The alternative is playing it safe and then hitting the enrage. It was really easy to get there, but you didn't win.
To go back to the idea that a Tank messing this stuff up has more severe repercussions than other roles, I can illustrate this with an example.
First, the Bad Tank. This Tank holds aggro and has the defenses to avoid dying, but he spends his days endlessly kiting monsters around, spinning them in circles, getting people cleaved, and turning his back to enemies whenever possible. This makes fights far more difficult and far longer than they ever needed to be and the members of this party will run far, far away from this Tank whenever they see him again.
In comparison, I would present one of the most famous examples of a DPS not fully utilizing their ability to ease the burden of a party: The Bard who does not sing. They do their DPS and everything falls pretty smoothly, but no songs. The other party members might identify that the healer could have used some MP in that one pull, or Foes might have been useful in this other one, but it's not too much of a bother because it didn't cause anything to explicitly go awry. It's possible that nobody will even notice that this Bard didn't sing because the dungeon/fight went smoothly in spite of it.
When a Tank messes these things up, it's irritating, it slows things down, it makes the game more difficult, and it gets people killed. If a DPS/Healer messes these things up, it just makes non-raid content slightly more difficult. A Tank who fails the duties I'm describing will not be wanted or effective in any content. A DPS or Healer who fails these duties will only be eschewed from raids because those actions otherwise will not be necessary. That is why they're important enough for Tanks that I named them as primary duties. For everyone else, they're not quite as "primary", so to speak.
These "extra jobs" are not given to the Tank by the community because of some perception of leadership. They're given to the Tank by the game itself because Tanks are deliberately designed to be the best at performing them. This is one of the major points of the Holy Trinity: every role is best at the things they are supposed to be doing. A Tank who isn't the leader still needs to be doing them.
Well, um, I apologize for the wall. I hope I have explained my thoughts adequately without seeming argumentative. If you don't agree, that's fine. Hopefully you at least know where I'm coming from now. Getting a factual answer on the "true duties of the Tank" would require speaking to the devs, and I don't know about you but I certainly cannot speak Japanese.