What is an "instance of damage"?

Question

A comment to my answer here proposes that each part of damage with different types is a separate instance. It matters a lot for weaknesses and resistances:
If a Brimorak deals 13 slashing, 4 evil and 3 fire damage with its flaming sword and a PC somehow has 5 resistance against slashing, evil and fire each, that means 15 damage if all of the above is one instance, but only 8 if it is 3 separate instances.

This is because of this part of the Resistance section:

If you have more than one type of resistance that would apply to the same instance of damage, use only the highest applicable resistance value.

So, how many damage instances is one hit from a Brimorak?

Answer 1

Each Damage Type is an Instance

The rules for dealing damage describe the process in four steps:

• Step 1: Roll the Damage Dice and Apply Modifiers, Bonuses, and Penalties
• Step 2: Determine the Damage Type
• Step 3: Apply the Target's Immunities, Weaknesses, and Resistances
• Step 4: If Damage Remains, Reduce the Target's Hit Points

Following these steps gives a few important facts about damage, mainly that it's determined by the damage dice + modifiers and has a single damage type. An attack like the brimorak's flaming sword would inherently be dealing three separate instances of damage through this process: 2d8+4 slashing, 1d6 evil, and 1d6 fire.

Resistances don't stack against any of these instances as you've quoted, so some hypothetical creature with 'resist physical 5' and 'resist slashing 10' would only apply the higher resistance to take 2d8-6 slashing damage. Neither would affect the evil or fire damage as different instances of damage.

Answer 2

It's unclear

'Instance of damage' appears trice on Archives of Nethys, in weakness, resistance, and mutagenic renovator. In none of these places is the meaning explained, and inferring that meaning from context leads to conflicting interpretations.

Weakness

If you have a weakness to something that doesn't normally deal damage, such as water, you take damage equal to the weakness value when touched or affected by it. If more than one weakness would apply to the same instance of damage, use only the highest applicable weakness value. This usually happens only when a monster is weak to both a type of physical damage and a given material.

Here the context suggests that there is an instance of damage for each damage type, because with this interpretation monsters "weak to both a type of physical damage and a given material" really are the only usual way in which two weaknesses apply to one instance.
Whereas, under the interpretation that one instance of damage can include several damage types there would be plenty of ways for multiple weaknesses to apply to one instance.

Resistance

If you have more than one type of resistance that would apply to the same instance of damage, use only the highest applicable resistance value.
It’s possible to have resistance to all damage. When an effect deals damage of multiple types and you have resistance to all damage, apply the resistance to each type of damage separately. If an attack would deal 7 slashing damage and 4 fire damage, resistance 5 to all damage would reduce the slashing damage to 2 and negate the fire damage entirely.

Here the context suggests that there shouldn't be an instance of damage for each damage type because that would make part about 'resistance to all damage' entirely redundant. Sometimes rules are restated redundantly in the CRB, but that doesn't usually happen literally in the following sentence.

Mutagenic Renovator

This sludgy concoction is said to be derived from liquefied mutant scales. For 1 hour after you imbibe the potion, your skin warps and mutates to grant you resistance 5 to one type of energy damage. When you first drink the potion, choose either acid, cold, electricity, fire, or sonic damage. The potion starts by granting you resistance against this type of damage. Each time you take damage from one of these listed energy types other than the one you currently resist, your skin mutates, causing you to lose the energy resistance previously granted by this potion and gain resistance to the type of energy by which you were most recently damaged, and the potion's duration decreases by 10 minutes. The resistance shifts only after you take the damage, so it doesn't apply to the first instance of damage.

Here the context strongly suggests that there shouldn't be an instance of damage for each damage type, because in that case you could have a weird behavior where a single effect with multiple instances of allows you to shift resistances more than once. However, the mutagenic renovator was published nearly three years after the rules for weakness and resistance, so I'm skeptical as to whether it's referecing the 'instance of damage' used in the CRB intentionally.

Even more context

More broadly, the rules for damage aren't particularly clear about multiple damage types.

• In step one, you roll and add the damage is together into one number.
• In step two, you need to determine the damage type for the damage you've calculated. How you're supposed to split up that number up for multiple damage types is kinda obvious in practice, but entirely unexplained in theory. And because the method to determine multiple damage types isn't clear, it's possible that this is also where multiple instances of damage come to be.
• In step three there's the ambiguity in weakness and resistance that I've already explained.

I would go so far as to say that steps one and two are very reticent on the notion of multiple damage types. Even the example in step 2 uses precision damage, which by its very nature increases "the attack's listed damage, using the same damage type", so doesn't really add another damage type.

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The ambiguity is confusing even to the developers that implement Pathfinder 2e for FoundryVTT, which automates the calculations for weaknesses and resistances.
For example, in March one such developer, nikolaj-a, stated on the FoundryVTT Discord:

I mean, the rules are a little vague on what constitutes an "instance" of damage. But it seems to be the term used for all the damage of a damage roll that is of the same type.

But then in April nikolaj-a also stated:

Well, no one seems to be willing to explain exactly what an "instance of damage" is 😛

So we're basically guessing, but in a way that makes it playable (mostly).