The rules themselves do not address this issue clearly, but the Sage Advice Compendium does... sort of
The following Q&A can be found in the Sage Advice Compendium:
Q. When a creature successfully saves against guardian of faith and takes 10 radiant damage, how much damage does that count against the total amount of damage the spell can deal? Is it 20 because that’s how much it dealt or 10 because that’s how much the target took?
A. It dealt 10 damage to the creature, so 10 is subtracted from the total.
Now, looking at the guardian of faith spell, it states:
[...] The creature takes 20 radiant damage on a failed save, or half as much damage on a successful one. The guardian vanishes when it has dealt a total of 60 damage [...]
And from this, we conclude almost everything we need.
Part of the question assumes that the target took 10 damage and that the attack dealt 20 damage:
Is it 20 because that’s how much it dealt or 10 because that’s how much the target took?
Meanwhile, the answer simply states that the guardian dealt 10 damage, meaning it did not deal 20 damage. It similarly never refutes the claim that the target actually took 10 damage, so we can assume this too to be the case.
And so, the guardian dealt 10 damage and the target took 10 damage, despite the halving that occurred. From these we can conclude that damage taken and damage dealt are equivalent quantities.
And yet, of course, there are problems
There are answers that argue these quantities must be different for certain features to make sense, and they are correct:
Resistance and then vulnerability are applied after all other modifiers to damage. For example, a creature has resistance to bludgeoning damage and is hit by an attack that deals 25 bludgeoning damage. The creature is also within a magical aura that reduces all damage by 5. The 25 damage is first reduced by 5 and then halved, so the creature takes 10 damage.
Whenever the golem is subjected to acid damage, it takes no damage and instead regains a number of hit points equal to the acid damage dealt.
From the first quote, one would note that it literally says the attack deals 25 damage and the target takes 10 damage. And from the second quote, if these were the same quantity then the Golem taking no damage would mean that no damage was dealt which means it would always regain zero hit points, which makes no sense. Thus, a non-zero amount of acid damage must have been dealt which is different from the damage the Golem takes.
I really have no way to argue against these claims other than that the books use Standard English and English is weird sometimes. I can find no way to reconcile "these quantities are always the same" with making that first quote and that feature ever making sense. You can treat them as separate quantities when needed, but the SAC treats that as the same quantity (at least in the case of guardian of faith).
I think my best argument for these quantities being the same thing is that if they were supposed to be different, that would have been explicitly stated, it wouldn't have been loosely alluded to in a few rules scattered throughout the books that only occasionally made use of the distinction that otherwise never came up.
And then the SAC makes it all more complicated...
There is another Q&A in the Sage Advice Compendium that that states:
Q. How does Arcane Ward interact with temporary hit points and damage resistance that an abjurer might have?
A. An Arcane Ward is not an extension of the wizard who creates it. It is a magical effect with its own hit points. Any temporary hit points, immunities, or resistances that the wizard has don’t apply to the ward.
The ward takes damage first. Any leftover damage is taken by the wizard and goes through the following game elements in order: (1) any relevant damage immunity, (2) any relevant damage resistance, (3) any temporary hit points, and (4) real hit points.
Now let's also look at the text of the Arcane Ward feature:
[...] Whenever you take damage, the ward takes the damage instead. [...]
Notably, this happens when you take damage, and from the SAC Q&A we can see that damage taken must be calculated before resistance is applied. But then what would we call the damage that happens after resistances are applied? Yeah... it's a mess and I do not believe the various features throughout these games were written with a level of precision that allows one to meaningfully analyze these terms in their entirety.
There just are edge cases and problems that arise under either interpretation (different quantities, or the same quantity), and resolving them is... unclear. I suppose, at the end of the day, the best you can do is ask your GM.
