When I promised @Charlaquin a separate post, I hadn't read yours yet!First, the obvious thing is that the 4E text is speaking about a particular mode of play that happens between encounters
To elaborate on what yuu say: 4e sets out exploration as something that happens between encounters. Encounters are defined as both combat and non-combat; the latter (per PHB p 9):
include deadly traps, difficult puzzles, and other obstacles to overcome. Sometimes you overcome noncombat encounters by using your character’s skills, sometimes you can defeat them with clever uses of magic, and sometimes you have to puzzle them out with nothing but your wits. Noncombat encounters also include social interactions, such as attempts to persuade, bargain with, or obtain information from a nonplayer character (NPC) controlled by the DM. Whenever you decide that your character wants to talk to a person or monster, it’s a noncombat encounter.
The topic of non-combat encounters is revisited on p 259:
Noncombat encounters focus on skills, utility powers, and your own wits (not your character’s), although sometimes attack powers can come in handy as well. Such encounters include dealing with traps and hazards, solving puzzles, and a broad category of situations called skill challenges.
When you (i) read these descriptions of non-combar encounters, and (ii) read the DMG on skill challenges which includes discussions of skill challenges nested within combat encounters, and (iii) read the DMG account of traps which allows that traps can occur within combat encouners, and (iv) realise that some skill checks (eg Atheltics to jump, Intimidate to get a foe to surrender, etc) may be made in combat, and (v) read the bit I quoted to Charlaquin which explains how skill challenges broadly follow the exploration loop, then I think the clarity of these categories - exploration, combat encounter, non-combat encounter - breaks down a bit.
And the remarks about adjudication - how the GM has to decide whether to call for a check, and what check to call for - clearly aren't confined to exploration. If a player in a 4e game has his/her PC, during a combat, move from A to B or lift up a rock or whatever, the GM has to decide if a check is needed and if so what sort of check (and the DMG has a whole discussion of this under headings like Hindering Terrain), as well as deciding what sort of action is required (Minor, by default, but that's not the only possibililty).
My point so far has been that, despite 4e's aspiration to classificatory clarity, it's actually all a bit of a mess as far as classification is concerned, and the exploration procedures very clearly bleed into the encounter part of play. But that doesn't change the fact that encounters are at the heart of play.
And this is where I see the contrast with the 5e rules text, which says - almost as an afterthought - that "In certain situations, particularly combat, the action is more structured". This is a reference to process, but carries with it no implication of situation, or finality of resolution, which for me at least is what is implied by encounter (although my reading of 4e here may be influenced by my knowledge of other parts of the rules text together with my long experience of how it actually plays).
This doesn't strike me as strongly as it does you, although I think I can see what you're seeing. (And perhaps what you're seeing here is what @Charlaquin was talking about in her post upthread.)the 5E text seems to vacillate between resolving actions by the DM decides and the dice decide, whereas the 4E text seems to do a better job of describing the relationship between those two things in a more concrete, procedural way.
Given that I am now seeing it (or at least seeing something in this neighbourhood), for me it seems closely linked to what I said just above about encounters. Encounters demand resolution; and resolution, or at least satisfying resolution, in a RPG demands a process that produces finality and isn't just more open-ended talking among participants! So it's not a surprise that 4e makes a better attempt at setting out clear resolution procedures. Because it needs them if its to deliver the game of encounters that it is promising. Whereas a game of exploration doesn't generate the same demand.
TL;DR: In 4e it is the encounter that is crucial; in 5e it is exploration that is crucial. Frankly, I think nearly everything else about the underlying cause of the "edition wars", about who does or doesn't enjoy 4e vs 5e, etc, flows from this.
(Maybe that's an exaggeration - probably some people played exploratory 4e, though I'm not quite sure how; and probably some play encounter-oriented 5e, though again I'm not sure how as the system will push against them. But I'm reasonably confident of my general thesis.)