The literal English meaning is the meaning of the descriptor of the DC. The experienced difficulty can diverge from the descriptor, as for instance a tier-3 character who could approach "hard" DCs with every confidence of overcoming them. That is why it I say it is unhelpful in the long run to look only from the low-level perspective represented by the descriptors.
It may not be helpful in all cases, but it
is what the designers decided upon.
Also, one could potentially argue the 80-20 rule here. ~80% of game play occurs at first tier so... yeah those DC
descriptors are spot on most of the time. (someone, please correct me if you have an accurate account of data to refute my 80-20 claim. The DnD Beyond character data are likely a reasonable approximation. Perhaps
more than half is a better estimate than 80%... but I digress).
Further, while certain difficult things
are easier for high level characters, the current paradigm of DC
descriptors (...10=easy, 15=medium, 20=hard...) still work for that individual 15th level Fighter who dumped INT and is now asked to make an INT(Investigation) ability check.
I believe that view is not correct or convenient. A DC is a fixed property of a task*. Let me preface by saying that I see this as an investigation, not a final position!
I appreciate the bolded reminder. I'm here to learn, too.
First from the point of view of rules - which is what we might label correctness (while acknowledging that correct is really whatever works for you at your table.) Approaches can change the likelihood of success or even obviate the need to make a check, but they do not change the DC of the task. In published material, such as ToA, numerous DCs are given by the designers. Nowhere does it suggest these DCs are formed on a one-to-one basis with actors. A DC 20 secret door in ToA is not DC 10 to one actor and DC 30 to another, but it may be much more probable for one actor to notice it over another. For instance, a character possessed by Papazotl, or one with Dungeoneer. The hard check doesn't become "easy" for one actor and "very hard" for another - where those are fixed descriptors - it remains "hard" i.e. DC 20.
Disagree. An ability check DC is very much dependent upon the approach taken by the PC in the context of said obstacle. Are you really going to say a locked door - which everyone in the party wishes to open - has the same DC for the rogue who is going to use their thieves tools vs the wizard casting
knock vs the barbarian who is going to run up and ram it with her body weight vs the bard who is going to punch it (he mad b/c the barbarian didn't like his song of rest)? The approach very much plays into the DC here. Or, let's simplify the example. Is it the same DC for the barbarian who is deciding between two options: 1. run up and slam her body weight into the locked door OR 2. whack at the lock with her great club?
Then there is the matter of convenience. It would be inconvenient to never know the DC until we knew the actor. It is far more convenient to know the DC of a task, and know that the actor may bring to bear an approach that changes their odds of overcoming it. As DM, I can know the DC is 20 without yet deciding the likelihood characters will have of overcoming it.
It's not about the actor, though. It's about the actor's approach and goal. In the previous example, if the barbarian and bard were both to throw their body weight into the door individually, the DC would be the same - just the barbarian might be more likely to succeed based on their stats and abilities.
Another example: there is a 15x15 foot pit blocking the way in a wider-than-usual dungeon corridor. Is the DM to assign a DC to the pit before finding out how the different PCs are going to approach this obstacle? IMO, that is backwards.
Note that the DMG pg 237-239 first instructs DMs to determine if the task proposed by the player is an auto-success or impossible for the PC. Then it tells DMs to only call for a roll if there is a meaningful consequence of failure. Only then, a full page later, does it get into setting DCs.
*It's not clear to me what distinction you are getting at by separating "obstacle" from "task".
Is the distinction clearer now?