Let me finish the quote of mine that you truncated... if a player saying they want their PC to climb the tower most of the way to get a good look around before summitting isn't an action and a goal, then I'm not sure what is. And perhaps that doesn't even matter because our job, as DMs, is not to tell the player what their actions and goals should be, it's to adjudicate them in the context of the scene. If the action and/or goal seem incomplete, of course the DM can - and should - prod for more. Clarity is at the heart of this playstyle. And please let's not mistake clarity for "magic words" as has been bandied about before.
As this is the D&D 5th Edition forum, of course we can assume it is a 5e game. Again, it is up to the player to describe their action and goal so there is no confusion - and the DM to prompt them if it is unclear. In combat, there is lots of short-hand going on. Attack rolls can assumed most of the time, not called for by the DM. Killing blows are assumed most of the time, so clarification is not necessary. It's the players responsibility to declare the desire for a knock out sometime during the combat - it's not on the DM to suggest courses of action for the PC in combat, or any other time, really. So, yeah, no awkward moment unless the player doesn't say how this attack is different. Keep in mind, we're not looking to set up "gotchas", we're looking for fluidity in game play and everyone at the table contributes to that. If a player retroactively says "Oh, I meant to knock it out not kill it" since they forgot to state the goal, well that's fine, it's just not fluid. This playstyle isn't about being a jerk DM as some like to paint it as.
The DM is in charge of narrating the results of the adventurers' actions - and as part of that, the DM can elect to have the player narrate instead, as in the case of taking down a foe. I don't see where this is prescribed that the DM must ask the player "how do you want to do this" after a successful blow that brings a monster to 0 HP - but perhaps I missed it and you can point it out. Regardless, it can get old if you are asking the same player how they used their sword differently this time to take down the 5th orc they've killed in the battle.
From last to first...
DND 5e rules in the PHB state
"Sometimes an attacker wants to incapacitate a foe, rather than deal a killing blow.
When an attacker reduces a creature to 0 hit points with a melee attack, the attacker can knock the creature out.
The attacker can make this choice the instant the damage is dealt. The creature falls unconscious and is stable."
That makes it clear that when a creature is dropped to zero by melee attacks the attack can choose to Ko instead of kill.
yes, the Gms job is to narrate the results but that does not mean skipping past choices the player can make that alter the outcome.
point is, it was the Gm choice to skip over the stage where the player can choose (by RAW) that led to your allegedly "awkward" case of the Gm jumping to the wrong conclusion. This is not a case of a player not stating his goal when he should, but of the Gm not asking when the choice was there for the player.
OBVIOUSLY if the table has agreed to a house rule which says these have to be pre-declared before each attack - the player cannot know it will be a drop-to-zero before the attack and so must declare every time - things change but thats the house rule issue.
But for 5e - the minimum extent of "how do you want to do this" is - "Are you going to KO?" when the zero hit occurs.
First case - sorry but the point made was the player could just as well had declared the distance they travel up the wall, where they stop and *not* what they intended to do once they got their.
its was an incomplete ACTION declaration - "how far do you climb" - not a lack of statement of whats in the mind of the character.
Like i said, if a player tells me their character walks down the corridor towards the T intersection - i need to know "how far" to resolve that action, not "what is he thinking about doing once he gets there." I cannot imagine circumstances in which i as Gm would "gotcha" him by having his character just walk out into the corridor and get spotted without him having given me more info.
It seems like the term goal here is being used more in the manner of describing the action and that the assumption really being put forth is that without that stated the Gm *will* move straight to a means in which that bites the character. yet at the same time its oh so greatly protested as "not gotcha" and "not magic words".
Me, i just show them in play and by simply asking "how far?" a few times early on if i need to.
But, in fact, i find that when the players-to-Gm conversation *includes* game mechanical info like "move 25'" or "making stealth check or using stealth" these examples are practically eliminated.
So, in my experience, your climb 25' up a 30' wall is resolved the same whether or not your goal is to peek over the top or to ferret out a item from the cliff thats only 25' up. You as a player told me how far you wanted to climb - knowing you do need to define "where" when you say "i move".
Again i come back to that the more one forces the players away from the mechanics, it seems the more they are opened up to getting these "GM decides how to resolve" that go against them - based on the examples provided.
But thats just me, perhaps.