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A number of features in the game enable a PC to reroll a d20 die as part of an attack roll, ability check, or saving throw. For example, the Halfling's Lucky feature and racial Bountiful Luck feat (emphasis mine):

Lucky. When you roll a 1 on an attack roll, ability check, or saving throw, you can reroll the die and must use the new roll.

Bountiful Luck. ...When an ally... rolls a 1 on the d20 for an attack roll, an ability check, or a saving throw, you can use your reaction to let the ally reroll the die...

However, a number of other features allow you to reroll an attack roll, ability check, or saving throw without mentioning the die. For example, the Path of the Zealot's Fanatical Focus feature or the Arcane Archer's Curving Shot feature:

Fanatical Focus. ...If you fail a saving throw while you're raging, you can reroll it [the saving throw], and you must use the new roll....

Curving Shot. ...When you make an attack roll with a magic arrow and miss, you can use a bonus action to reroll the attack roll against a different target.

What's the distinction between these two kinds of wording?

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    \$\begingroup\$ As this is about D&D 5e, there's no need for the other two tags (D&D in general would be when the question isn't about a particular edition and d20 System is a different game). So I've removed those two tags. \$\endgroup\$ – Bloodcinder Aug 18 '18 at 4:05
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    \$\begingroup\$ It's unclear what you're asking. Are you trying to clarify the distinction between features that specifically allow a reroll on the d20 as part of an attack, saving throw, or ability check rather than a reroll on the attack, saving throw, or ability check itself? In other words, is the distinction whether features mention the d20 die or not? \$\endgroup\$ – Bloodcinder Aug 18 '18 at 4:12
  • \$\begingroup\$ @Bloodcinder yes \$\endgroup\$ – illustro Aug 18 '18 at 7:59
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    \$\begingroup\$ I edited the title and text of your question based on that "yes" answer to my previous comment. I also included examples with emphasis so people can see exactly the difference you're talking about. \$\endgroup\$ – Bloodcinder Aug 18 '18 at 12:49
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Mechanically they end with the same result, rerolling the d20, the difference is when they can be triggered

There will be specific differences between a lot of these will be in the detail of the ability/feature/feat is being invoked, and in those situations the specific beats general rule will modify what is laid out here.

D&D is a d20 game where the primary mechanic is rolling a d20 dice.

Mechanically, a vanilla saving throw/ability check/attack roll (ie one without advantage/disadvantage) has this Order of Operations (OoO):

  1. Roll d20
  2. Add modifiers
    • These modifiers can be in the form of static modifiers (like the Constituation modifier) or variable modifiers (like the d4 that Guidance provides for an ability check)
  3. Work out the total of the roll
  4. Declare it to the DM
  5. Find out if it succeeds/fails

(See PHB pg 7 D20 section for a more in depth discussion on the d20)

The primary difference between "reroll a (d20) dice" and "reroll an attack roll/saving throw/ability check" is when the trigger for being able to activate that ability occurs.

Reroll a (d20) dice trigger

This ability (in general) can be activated after step #1 but before step #2 in the above OoO. For example, the Halfling's Lucky feature triggers when you roll a 1 on a d20 dice.

Reroll an attack roll/saving throw/ability check

In general the trigger for this ability to activate will occur after step #5 in the above OoO (depending on the specific wording). For example the Path of the Zealot's Fanatical Focus can be triggered when you fail a saving throw (ie after step #5).

But what happens when the ability is triggered?

In both cases the same thing happens, you start a new OoO, with a new step 1. and continue from there. The specific ability may require you to use the result of this new operation.

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Generally, the only difference is in the eligibility to reroll.

In the examples of the Halfling's Lucky and Bountiful Luck features, the reroll is only eligible to be performed when the d20 itself resulted in a natural 1, regardless of the total for the roll including modifiers or the success/failure outcome of the roll. This restriction on the natural result is typical of features that allow rerolling and mention the actual d20 die.

The wording avoids the ambiguity of conflating the natural die result with the total result of the attack, check, or save. For example, see the Rogue's Reliable Talent (which does not involve actually rerolling the die but provides a similar benefif of pretending the die rolled a particular number):

Reliable Talent. ...Whenever you make an ability check that lets you add your proficiency bonus, you can treat a d20 roll of 9 or lower as a 10.

The wording makes it clear that you get to pretend you rolled a 10 on the die when the die itself resulted in a 9 or lower, not when the total including your ability check modifier was 9 or lower.

By contrast, in the examples of the Zealot and Arcane Archer features, the reroll is only eligible to be performed after resolving the attack, check, or save to determine that it failed. This restriction on the outcome of the roll all-things-considered is typical of features that allow rerolling but don't mention the actual d20 die.

In either wording, there is no practical difference in what you do as a player and how you benefit from it: you still reroll the d20 die to possibly change the total result and the outcome. Only the eligibility condition involved differs: typically you are eligible on a particular natural result when the die is mentioned and on a failure or other specific outcome when it is not.

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