Basically, we are talking about how much the DM should have the ability to control what the players can do and how the players can react to a situation.
As a DM, I never control what the players do, or how they react. If a player wants to leap a 200-foot gap, they are welcome to try. I supply the players info and they decide their actions. What I will do is assign odds and create spot rules to cover what the rules don't cover. (Something that proved a lot faster than the endless "look it up in the book" in the 3e days.) There's a world of difference in creating a mechanic or assigning odds versus controlling what a PC does.
I suppose I occasionally "control" a PC's action by making a task so difficult as to be almost impossible. If a player wanted to leap the aforementioned 200-foot gap, my brain tells me that the real life broad jump record is something like 30 feet, so they'd have a hard time of it. So I might assign a miniscule chance of success (something the player would usually be warned about). But my players would use common sense and it's doubtful the topic of jumping such a span without magical aid would come up in the first place.
Anyways, there really at least three different ways I can think of that this situation could be approached.
1) The DM arbitrates player actions. The DM controls all aspects of the situation, and any action by a PC requires either information from the DM or explicit permission by the DM. This need is generally implicit in the rules rather than explicit, but it is still there. This is what I would call the old-school approach that is often derided as "Mother May I?".
2) The rules arbitrate player actions. The situation is formally controlled by the rules whenever possible, with only unusual situations falling under DM control. This kind of game tends to use things like a battle grid, strong and complete rules, and strongly codified player abilities. This is the 4E-style approach that is often derided as "boardgamey" or "videogamey".
3) The players arbitrate player actions. The situation is equally controlled by everyone at the table, and everyone has a clear ability to create elements of the setting and situation to suit their needs. Players don't need to ask the DM if a chandelier exists because they are free to write the chandelier into the setting themselves. This is more the indie RPG approach that is derided as a "storygame, not a real RPG".
This is well stated. But I never find play as absolute or rigid as this, but rather a mix of 1 and 2. As DM, playing 1e or 2e, I would not say that "any action by a PC requires either information from the DM or explicit permission by the DM." Not by a long shot. Ninety to 95% of the time the players take routine actions, attacking, spellcasting, etc., based on my initial description. My experience would probably be most accurately described by number 2 above, even when I was playing early editions.
For instance, I carefully describe a cave. The players typically ask a few questions about the finer details—something that happens in my 4e campaign too—and then they state what they wish to do. If the cleric wants to cast a Light spell, and the mage wants to sit down and examine the scroll found in the last cave, and the fighter wants to rest and polish his armor, they can do that without expecting permission or much further detail from me (unless unexpected visitors show up). If troglodytes suddenly arrive, everyone rolls initiative and the fighter can run to the attack, the mage cast Sleep, etc., again without explicit permission or detail being required.
The DM caveat (or whatever you want to call it) tends to show up in those odder circumstances, such as the fighter wanting to throw his sword so it slices through a chandelier rope or the thief wants to dangle from a rope and disarm a trap one-handed.
In the end I don't think our approaches are all that different in practice. I think what I'd like to see preserved/restored in future game editions is the traditional style of DM-player interaction.
1e style:
DM: You've entered a dusty chamber with a vaulted ceiling... (yada yada)
Player: I'm going to look around for anything of value.
DM: Whereabouts are you searching?
Player: I'll check the room corners, brushing away dust as needed.
DM: Nothing there but some old cobwebs, and a few tiny, yellowed bones in the northeast corner—possibly the remains of a rat.
Player: How about higher up? I look up at the ceiling beams.
DM: Roll percentile dice for me.
[Player rolls.] Player: I got a 87.
DM: You discover an old coffer tucked away on rafter.
4e style:
DM: You've entered a dusty chamber with a vaulted ceiling... (yada yada)
Player: I'll make a Perception check. [Rolls die.] I got a 24, which includes my racial bonus and Soaring Eagle Eye utility power.
DM: You discover an old coffer tucked away on rafter.
Now the second example certainly cuts to the chase quicker. But to me, it's devoid of soul or feeling; comparing the two is like putting poetry next to a car repair manual. Yet the latter approach seems to be what some people are advocating—"Rules exist for passive and active Perception, so let me call the skill and make my roll, and let's not bother with the details."