Grammatically speaking, there is a very important difference, and that is order of operations.

When you are reduced/drop to 0 hitpoints

The order of operations is as follows:

1. You take damage
2. You have 0 hit points
3. The feature that activates when you are reduced to 0 HP kicks in
4. You have 1 HP instead

When you would be reduced/drop to 0 hitpoints

The order of operations is as follows

1. You take damage
2. The damage would have reduced you to 0 hp
3. You have 1 HP instead

Thus, the big difference here is that features that activate when you are reduced to 0 hitpoints mean you still had 0 hitpoints for a moment. On the other hand, features that activate when you would be reduced to 0 hitpoints activate before you hit 0 HP, so you never had 0 HP.

This is important for effects that immediately kill you if you hit 0 hitpoints. There is a slightly out-of-date sage advice comment to this effect:

If the damage from disintegrate reduces a half-orc to 0 hit points, can Relentless Endurance prevent the orc from turning to ash? If disintegrate reduces you to 0 hit points, you’re killed outright, as you turn to dust. If you’re a half-orc, Relentless Endurance can’t save you.

I will note (as an aside) that since this Sage Advice item was created, errata has altered the Disintegrate spell--which likely changes the outcome laid out for that specific spell. (That bit of Sage Advice is from 2016, the errata is 2018)

The target is disintegrated if this damage leaves it with 0 hit points.

Based off of this change in wording, effects that activate when you are reduced to 0 HP would still save you from Disintegrate because you are not left with 0 hit points. On the other hand, a Beholder's Death Ray...

The target dies if the ray reduces it to 0 hit points.

In this case, an effect that activates when you are reduced to 0 HP would not save you--because you were reduced to 0 HP before that feature activated, and thus died before the effect could activate. However, an effect that activates when you would be reduced to 0 HP can save you--because you were never actually reduced to 0 hp.

As a bit of non-official support, here's a comment from J. Crawford on Twitter...

Q: does death ward prevent disintegration?

A: Death ward works against any effect that would reduce you to 0 hp via damage or that would kill you instantly without dealing damage.

While Crawford's tweets are no longer considered official, it does offer perspective. Back when Disintegrate could kill you through Polymorph or Relentless Endurance, Death Ward could protect you from those before you never actually reached 0HP.

Order of Precedence

This is a bit of odd English to try to explain, but I'll do my best. The sequence in which the four terms used here would happen in goes like this:

1. Would Be: This is future-facing and happens first, in anticipation of the event
2. Reduces: Active Event, this occurs simultaneous to the triggering event
3. Are Reduced: Reacts to the event, this occurs after the reduction has happened
4. Leaves it with: Final state, occurs once all 'events' have ceased.

TL;DR:

A Half-Orc's Relentless Endurance would not save you from death via being reduced to 0 HP by a Beholder's Death Ray--while the spell Death Ward would. This is because Relentless Endurance activates after you have been reduced to 0 HP, while Death Ward happens before you would be reduced to 0 hp.

The d3 has been used for many different mechanics.

This search on DnDBeyond gives all the instances of "d3" appearing in the DDB compendium material, which covers the first three sections here. Note, this one has several false positives, as "D3" is used to denote locations in many adventures. All the other searches I link here use "1d3" to get hits.

The d3 first shows up in the introduction to the Player's Handbook, where it is explained how you can simulate one with a d6:

The same d notation appears in the expressions “1d3” and “1d2.” To simulate the roll of 1d3, roll a d6 and divide the number rolled by 2 (round up).

We now examine many of the ways the d3 has been used.

Falling down a chimney.

In one of the adventures published in Explorer's Guide to Wildemount, you can fall down a chimney:

Any character who tries to descend the chimney without a rope or a climbing speed inevitably falls. Roll a d3 to determine which lower chimney the character tumbles down.

Betting on dinosaur racing and meeting with princes.

In Tomb of Annihilation, you can bet on dinosaur races at one of the major towns. Two of the possible results depend on the outcome of a d3 roll.

In this same town, the party can seek an audience with a merchant prince. The module states:

Characters must wait 1d3 hours before being given an audience.

Having close family relationships.

In the character building options in Explorer's Guide to Wildemount, you roll a d3 to determine how many of your family members you are close to:

Roll a d3. This is the number of powerful relationships you have within your family.

And a plethora of magic items.

This search shows 33 magic items that use a d3 in some manner, most of them are for recharging at dawn. I will not list them all here.

The Ioun Stones use a d3 to determine how far from your head the stone orbits.

The decks of Many Things and Several Things use a d3 to determine the number of times a certain ability can be used.

And a few monsters.

This search gets us the mosnters. Five monsters use a d3 to determine the number of creatures they summon with a particular ability. For example, the Crokek'toeck:

Crokek’toeck opens its mouth and disgorges 1d4 barlguras, 3d6 gnolls led by 1 gnoll fang of Yeenoghu, 6d6 dretches, or 1d3 vrocks.

The Assassin Bug lays eggs inside you:

The target is infested with 1d3 assassin bug eggs, which immediately hatch into assassin bug maggots.

The Olhydra makes it rain:

Violent downpours become frequent within 10 miles of the lair. A downpour occurs once every 2d12 hours, and lasts 1d3 hours.

How many slaves does a Booyahg Goblin have?

This search gets us the goblin playable race. But this is misleading. DDB took the Monster Lore section on Goblins from Volo's Guide to Monsters and put it on the same page with the goblin playable race information. I recommend against giving your players slaves.

Booyahg Whip. Khurgorbaeyag saw fit to gift this goblin with powers that enable it to dominate others. The goblin has 1d3 other goblins that slavishly obey its orders.

And an indeterminate amount of times for random encounters.

This search gives an overflow of results at DnDBeyond. I'm not going to count them. Most are random encounters from Xanathar's Guide to Everything, but there was a hit for Tales from the Yawning Portal, as well as one for Icewind Dale: Rime of the Frostmaiden. .