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?