DM Page 238: It's your job to establish the Difficulty Class for an ability check or a saving throw when a rule or an adventure doesn't give one. Sometimes you'll even want to change such established DCs. When you do so, think of how difficult the task is and then then pick an associated DC from the Typical DCs table.
Nowhere does it say DCs are objectively meaningful. If I establish that a check is easier for one person than another for some reason, that is my option.
If the players felt I were doing this to be a jerk DM, they will let me know. If I'm doing it to highlight someone's character or give them a chance to do something extraordinary, they don't have much of a problem with it. I believe that is exactly what the 5E skill system is intended for.
It also does not say to establish multiple DC's for the same check. A lock is a lock is a lock. Note, you are the only one who is talking about this being a problem at the table. You are perfectly free to do what you want. But, you are not free to pretend that the rules support your interpretation in any real way. They don't. So, when I call you on this, it's not that I think you're being a jerk or doing anything wrong. Just that the rules don't actually allow for this. It's a perfectly fine house rule and it apparently works for you. Great.
If your eyeballed estimate for the volume of a paraboloid object happens to be within 1% of the value determined by my application of calculus, why am I able to derive extra information that you couldn't?
Because training matters.
But, we made the same check and got the same information - the volume of a paraboloid object. Any additional information would require a separate check. Or for the two of us to have gotten different numerical results from the same check. I've got no problem with one character getting better results from a better check. That makes sense. OTOH, if I, untrained, score higher than you, I should also get more information. Training simply doesn't mean that much, particularly when we're talking about the skill checks being made in D&D.
Again, if your ultra trained ninja scores a modified 20 stealth check, and my character gets lucky and untrained with a 10 Dex scores a 20 as well, then you aren't harder to detect than I am. We are both, in this case, identical. Now, most of the time, you're going to score far higher than me, so fair enough. But, why give individual results for one kind of skill check and not another? It's not consistent. When does training matter? How often does training matter? Is it only Int based skills? Does my training in Persuasion mean that I get better results than your high Cha character? What does that actually mean in play? Does training in Deception mean that my lies are harder to detect even though the untrained character got the same result?
Like I said, this is a very deep dark rabbit hole to climb into.
No, no it isn't possible. 2d6+3 for the magical greatsword has a minimum damage of 5. 1d4+0 for your dagger gives a maximum of 4. You could never under my example do equal damage with an equal roll.
Ahh, my bad. I was thinking Greatsword was D10.

I'd point out though, that training still has nothing to do with damage. A trained dagger expert does the same damage as an untrained dagger user. A 20th level fighter does the same damage with that great sword as a peasant. Training has zero impact on damage. Your example falls apart because you are not comparing like to like. The great sword does more damage than a dagger, sure, but that's a result of the nature of the weapon, and has exactly zero to do with the user.
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Yes to all of those. People trained in climbing and jumping will kick my rear at those things.[/QUOTE]
In game? No, they won't. Training means nothing in those examples. Your distance jumped depends solely on the end result, not how you got there. So long as the character beats the DC, there is zero difference in how that character gets up that wall.