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All attack skills deal attack damage, but some of them will also deal non-attack damage.
There are four kinds of damage in PoE:
They have no overlap, no damage is more than one of those things.
So Prodigious Defence applies to attack damage specifically, which is damage from hits that is based on your weapon.
Advance Guard applies to any damage dealt with an attack skill, even if that damage isn't attack damage - it could be secondary damage (the corpse explosion on infernal blow, which is based on the enemy's life, not your weapon), or damage over time (such as if the attack ignites an enemy). Both of those would be examples of damage that isn't attack damage but is dealt by an attack skill.
Similarly, not all damage dealt by a spell is spell damage - damage over time isn't spell damage, and spells that blow up corpses use secondary damage for that.
Attack damage is any source of attacking including default attack, attack skills are skills like dual strike, heavy strike, etc
Default Attack is an attack skill.
The default attack actually is a skill.
In this case, Advance Guard will buff damage over time caused by attack skills, whereas Prodigious Defence will not.
I can't think of any cases where Prodigious Defence is better than Advance Guard, since I can't think of any ways to deal attack damage besides through the use of an attack skill. Or is there something that allows attack damage to apply to something that isn't an attack? Hmm.
Or is there something that allows attack damage to apply to something that isn't an attack? Hmm.
There is not (and for technical reasons quite probably never will be). Stuff like making spell damage modifiers apply to attacks is straightforward, going the other way is not, in general.
For clarification, is it always true that any hit whose base damage is not a damage number listed on the gem, or a damage number listed on your weapon, secondary damage? Or is that just a rule of thumb?
(Shield Charge and Spectral Shield Throw immediately come to mind as semi-exceptions, but they override "off-hand damage" which then presumably counts as attack damage.)
Are Attack and Spell damage always mutually exclusive, when looking at different components of a single skill? Is there any skill that deals both?
For clarification, is it always true that any hit whose base damage is not a damage number listed on the gem, or a damage number listed on your weapon, secondary damage? Or is that just a rule of thumb?
(Shield Charge and Spectral Shield Throw immediately come to mind as semi-exceptions, but they override "off-hand damage" which then presumably counts as attack damage.)
I should probably have worded that in terms of Main/Off hand damage - you are correct that the shield skills deal attack damage.
Outside that case, I believe this is currently true, but not necessarily a rule. A skill could have an amount of base secondary damage on it's gem to use for secondary damage hits, in theory. Explosive Arrow's explosion before it was reworked used to be secondary damage that had an amount per arrow on the gem.
It seems unlikely we'd do that for a spell (it would seem to serve no purpose), but the base value being on the gem is something that is (at least mostly) true of spell damage, but is not necessarily exclusive to only spell damage.
Are Attack and Spell damage always mutually exclusive, when looking at different components of a single skill? Is there any skill that deals both?
There are no skills that deal both attack and spell damage (and to be clear, no damage can be both at the same time). Nothing really stops us from making a skill that deals one hit of spell damage and one hit of attack damage, but that would be super weird and I don't think we'd ever do it.