Players don't declare actions, they declare intent. The GM decides if that intent translates into a corresponding action and what the outcome of that action is.
Says who?
Here is the quotation from Gygax's PHB (p 22) and DMG (p ), which makes it clear that the player of a paladin of 4th level or up can declare the action of praying for a warhorse:
At 4th level - or at any time thereafter - the paladin may call for his warhorse . . . it will magically appear . . .
When the paladin reaches 4th or higher level, he or she will eventually call for a warhorse (as detailed in the PLAYERS HANDBOOK). It will magically appear, but not in actual physical form. The paladin will magically “see” his or her faithful destrier in whatever locale it is currently in . . .
The fact that the fictional situation has to be appropriate - you mention a loud noise, and I think most tables would accept that you can't pray for your warhorse in the middle of battle, though I can imagine a group thinking differently - doesn't mean that the player can't declare the action. If my PC is not carrying a sword or other bladed weapon then I probably can't declare the action "I draw my blade and cut him in two!" - but that doesn't show that everything is gated behind the GM's permission, only that permissible action declarations depend upon the fictional position of the PC. Which is a largely uncontroversial point.
Due to the very nature of the RPGs, the GM has broad discretion with regard to what's happening.
This claim is too general to be very useful in describing different approaches to RPGing, and is certainly unhelpful in response to the OP's specific question about technique.
Just sticking to the example of AD&D combat rules, because they are pretty familiar to most posters, there are all sorts of things which
happen but are not subject to the GM's broad discretion: parameters for movement (set by a combination of movement rates and rules for being engaged in melee); who hits whome (determined by to-hit numbers and dice rolls); hit point totals (determined by pre-established starting states plus deductions that result from application of the to hit and damage rules and (less frequently) increases that result from application of the healing rules); etc.
In Classic Traveller (a game from 1977, so a fairly early RPG), the standard way to determine the response of a NPC to some request or approach by a PC is to roll on a reaction table. There are rules for modifying that roll. There are also discrete subsystems that apply to interactions with officials (mediated through the rules for Bribery and Admin skill). But there is no general assumption that the GM has broad discretion to decide all NPC responses to PC requests and approaches.
And to return to the example of "I draw my blade and cut him down!", in many RPGs the key fictional positioning requirement for that action declaration - namely, that the PC be carrying a blade - is under the player's control, not the GM's, because in many RPGs it is the player who chooses his/her PC's equipment (within system-determined parameters).
Obviously the examples of RPGs circumstances in which the GM does not have 'broad discretion with regard to what's happening" could be multiplied extensively.
Under normal circumstances, I expect a GM simply to determine what's going to happen based on his understanding of the game world.
That's an important fact about you and your RPGing preferences. It helps you work out which systems you might or might not enjoy, and who would be a good or bad GM for you. But again, it doesn't express any sort of general truth about GMing and RPGing techniques.
the GM sometimes needs to (subtly) steer the flow of the game. If the players are endlessly stuck in an investigation, then it might be alright to help them out. In such a case, a GM in the above situation might make a covert d100 roll, disregard the results, and declare a member of super-secret organization present any way - to advance the plot.
Again, this is one way of RPGing but not the only one, and I would suggest didn't become widespread until the mid-80s (and seems to have dominated since then).
But of course there are other approaches that are well known and are supported by a variety of systems. For instance, in a system in which the players can declare an action to search for sect members at various tea houses (Traveller is such a system, using the Steeetwise skill) all the GM has to do is adjudicate those atttempts, by setting difficulties and then - if the check fails - establishing what, if anything, happens as a consequence of failure. (Obviously if the check succeeds, the PCs find what they want.) There is no need to "steer" things, subtly or otherwise.
The idea that the GM should deceive the players about what action resolution system is being used - eg in your example, the GM pretends that the resolution is probabilistic but in fact the GM is just deciding - is quite controversial among RPGers, at least in my experience. For my part, if I am going to "say 'yes'" rather than call for a roll of the dice (most often that would be because there is nothing dramatic at stake in the outcome, and hence no obvious dramatic consequence of failure) I don't pretend otherwise. I just tell the players that their PCs get what they want, and then play continues on. An example is likely to come up in my next session of Traveller: one of the PCs needs medical treatment for an otherwise fatal disease (currently the PC is in cryogenic storage), and when the players say that they look for a hospital to take him to, I will say "yes" straight away and try to move as quicly as the table dynamic permits to events that occur once he has recoevered.
Or simply put: if you're too obvious and too transparent in manipulating game world objects and events towards a given story purpose, it might come across as too convenient to your players. You can do that but a light hand would be well advised, if possible.
Is this based on your own experience? Or are you just conjecturing?
In my experience, if the players say (for instance) "OK, now that we're back in town we buy such-and-such gear" and I reply "OK, no worries, write it down and mark of the right amount of money" that does not cause any problems. My experience similarly leads me to believe that my proposed treatment of the "We take him to hospital" action declaration that I describe above will not cause any issues at my table.
to advance the plot.
<snip>
to ensure the game remains on-track
Many RPGers do not play games in which there is such a thing as "the plot" or "remaining on track". In talking about the GM's role in such terms you are already focusing on a very particular (if quite popular) approach to RPGing - roughly speaking, an approach in which the GM narrates the players through a pre-authored story using a mixture of mechanical action resolution, overt GM decision-making, and covert GM decision-making of the sort you described in my preceding quote from your post.