Cutting Words applies after resistance because it is not damage reduction, it is subtraction with unique and explicit behaviour.
Consider the text of Cutting Words:
When a creature that you can see within 60 feet of you makes an attack roll, an ability check, or a damage roll, you can use your reaction to expend one of your uses of Bardic Inspiration, rolling a Bardic Inspiration die and subtracting the number rolled from the creature’s roll. You can choose to use this feature after the creature makes its roll, but before the DM determines whether the attack roll or ability check succeeds or fails, or before the creature deals its damage.
The term subtract is incredibly rare in DnD5e. It is used 9 times in the entire basic rules. It is only ever used when referring to the mathematical operation;
- "For disadvantage, subtract 5 [from a passive skill check]"
- "To determine an ability modifier without consulting the table,
subtract 10 from the ability score and then divide the result by 2
- "Whenever a creature takes damage, that damage is
subtracted from its hit points.", etc.
The term "subtract" should not be understood as the gameplay mechanic reduce damage.
With all other uses of the feature the subtraction happens "before the DM determines whether the [roll] succeeds or fails". This is after all modifications and other effects have been applied, right before the DM resolves the roll.
I will argue that this same intent--"before the DM resolves the roll"--also applies to damage rolls which is why they chose the unique term "subtracting" to distance it from existing mechanics and prevent confusion. The timing of the skill, "before the creature deals its damage", is explicitly mentioned. This is unusual because if this was a normal 'damage reduction' then the timing is already defined in the Combat rules, and this is not where it would happen. Normal 'damage reduction' happens "before resistance and weakness is applied".
What exactly does "deals its damage" mean? This isn't a normal choice of words. This phrase is used in a lot of contexts but it's not really clear to me. "Swords deal slashing damage", "an attack deals 5 damage".
The rules for Object Armor Class provides a hint:
An object's Armor Class is a measure of how difficult it is to deal damage to the object when striking it (because the object has no chance of dodging out of the way).
This suggests "dealing damage" does not mean rolling AC, which is a start.
Knocking A Creature Out says:
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.
This seems to say that "damage is dealt" is when the damage is applied. That is to say, after rolling damage you can choose to knock the creature out, you don't do it before rolling damage.
In most circumstances resistances are the last thing to be resolved. However, the specific overrules the general, and the text "before the creature deals its damage" can be interpreted as "directly before the creature deals (ie reduces the hp of the target) its damage". As opposed to if the text read "after damage is rolled".
In the rules for Making An Attack:
Choose a target. Pick a target within your attack's range: a creature, an object, or a location.
Determine modifiers. The DM determines whether the target has cover and whether you have advantage or disadvantage against the target. In addition, spells, special abilities, and other effects can apply penalties or bonuses to your attack roll.
Resolve the attack. You make the attack roll. On a hit, you roll damage, unless the particular attack has rules that specify otherwise. Some attacks cause special effects in addition to or instead of damage.
The DM deals the damage at the end of step 3, after all resistance and reduction is applied. Cutting Words is a special case subtraction that does not follow the normal rules, which is why it has special, unique wording. It is intended to modify the damage roll directly before Step 3.
This is just one reading, but it would resolve the question of "can I pick what type of damage to reduce". It also resolves associated questions like "does the troll regenerate", and a host of other small problems with picking or not picking what damage type you want to reduce.
- While this answer is RAW, there is an assumption that the designers decided to use the word "subtraction" for a specific reason, and that they chose to explain the mechanics because they are abnormal. It is possible that the designers simply chose the word "subtraction" offhand. However I find it more useful to assume the designers are intentional with their language.
- Sometimes skills do repeat core mechanics, and it's possible the designer intended the description to be a general overview not actual mechanics, or they accidentally included mechanics from an older test version and it just slipped through the cracks and made it to publishing. I don't think this is a good argument as it brings all rules into question.
- Spirit Shield doesn't use this language, and so it is another issue entirely.
- There is apparently some debate as to whether "taking damage" and "being dealt damage" are the same thing. I think they are, and I have included arguments in support of this above.
In the comments Medix2 asked: "if an attack does, say, fire and cold damage, which ones are reduced by Cutting Words?"
The answer is that Cutting Words subtracts from the damage roll, not from any specific type. Follow the steps as above;
- Roll the die
- Apply all modifiers
- Apply all resistances or weaknesses
You can now subtract the number rolled from the creature's attack. At this point the rules run out, all damage is assumed to be amalgamated and subtracted from the creature's health. Cutting Words is a special case that is not clearly covered by the rules.
This is important in the case of a Troll, it can only regenerate if not dealt fire/acid damage.
Your choices are:
- Let the DM decide what damage has/has not been dealt (rule in favor of the monster)
- Let the Bard decide what damage has/has not been subtracted (rule in favor of the players)
- Ignore the question and have all damage types that get to this stage be 'dealt' but accept that it is possible they are 0 damage (unless the total damage is reduced to 0, then it would be reasonable to say that the creature takes no damage of any type).
- Some other option, again this is not covered by the rules so pick whatever you think makes sense or makes the narrative interesting.