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[majiponi]

When do we pay card costs?  For example, when ...
a Haggler is in-play,
I bought Innovation,
I earned $8,
I buy a Fortune,
I gain a Black Market, playing immediately,

can I buy the revealed King's Castle, spending what coin I have, fail to pay $8 (Fortune), and take 8 debts?  The answer will definitely be no, but nobody has denied the paying is one of the on-buy effects.

Another case:
a Haggler is in-play,
I bought Innovation,
I earned $4,
I buy a Herald,
I gain a Vassal, playing immediately,

can I overpay $2?

[GendoIkari]

When do we pay card costs?  For example, when ...
a Haggler is in-play,
I bought Innovation,
I earned $8,
I buy a Fortune,
I gain a Black Market, playing immediately,

can I buy the revealed King's Castle, spending what coin I have, fail to pay $8 (Fortune), and take 8 debts?  The answer will definitely be no, but nobody has denied the paying is one of the on-buy effects.

Paying is not a separate step that is triggered by buying. Paying is part of what defines buying. From the base game rulebook:

Your total amount of coins available to spend goes down by the cost of the card.

That's just a thing that automatically happens as part of the act of buying a card. By the time any "when-buy" things trigger, you've already lost the coins / spent them / paid for the card. So no, you cannot do the above because paying is not an on-buy effect.

Another case:
a Haggler is in-play,
I bought Innovation,
I earned $4,
I buy a Herald,
I gain a Vassal, playing immediately,

can I overpay $2?

Yes, this you can do. Because unlike paying, over-paying IS an on-buy effect. It's right there in the wording itself: "When you buy this, you may..." "Overpay" is really just shorthand for "pay some amount of and/or ". You aren't literally overpaying for the card, you already paid exactly what the card costs, and now you are paying some additional amount to trigger an additional effect.

Though no input from Donald, that understanding of Overpay is confirmed here: http://forum.dominionstrategy.com/index.php?topic=8709.msg264177#msg264177

And then there's this also, in which I asked the same question again 2 years later, apparently completely forgetting how it worked.

[Donald X.]

Overpay is in fact a when-buy trigger. It gets ordered with other when-buy triggers.

[GendoIkari]

Overpay is in fact a when-buy trigger. It gets ordered with other when-buy triggers.

Was this something that was intended when creating the cards; or simply a ruling based on how it needs to be that way because of how it was worded?

[Wizard_Amul]

Overpay is in fact a when-buy trigger. It gets ordered with other when-buy triggers.

Was this something that was intended when creating the cards; or simply a ruling based on how it needs to be that way because of how it was worded?

I don't know if the Guilds rules changed in different versions, but the Guilds expansion rules document on RGG does say "Overpaying happens when a card is bought, which is before it is gained. Players can only overpay for a card when buying it, not when gaining it some other way." So, "when buying" is explicitly stated in the rules, although the whole idea of the overpay being like a separate event isn't explicitly stated.

[Donald X.]

Overpay is in fact a when-buy trigger. It gets ordered with other when-buy triggers.

Was this something that was intended when creating the cards; or simply a ruling based on how it needs to be that way because of how it was worded?

It was always a when-buy trigger; it wasn't a ruling later.

[majiponi]

Overpay is in fact a when-buy trigger. It gets ordered with other when-buy triggers.

Was this something that was intended when creating the cards; or simply a ruling based on how it needs to be that way because of how it was worded?

It was always a when-buy trigger; it wasn't a ruling later.

I cannot see the conclusion.  You mean, overpaying should be done with ordinary paying, which spends coins (or potions) and Buys, and triggers on-buy effects and gaining what we bought?

[GendoIkari]

Overpay is in fact a when-buy trigger. It gets ordered with other when-buy triggers.

Was this something that was intended when creating the cards; or simply a ruling based on how it needs to be that way because of how it was worded?

It was always a when-buy trigger; it wasn't a ruling later.

I cannot see the conclusion.  You mean, overpaying should be done with ordinary paying, which spends coins (or potions) and Buys, and triggers on-buy effects and gaining what we bought?

Overpaying is a “when-buy” ability, and “when-x” abilities always happen after you finish doing x. So overpaying is not done at the same time as paying; it’s a completely separate thing from buying the card; a special ability that happens after you finish buying a card; just like Haggler.