Spell Damage + CoE + HoA interaction

Spell Damage is a method of dealing damage. There are, as has already been pointed out, four of these: Spell Damage, Attack Damage, Damage Over Time and Secondary Damage. All damage dealt is in one (and only one) of these categories.

From the 1.0.2 Patch Notes (I've bolded the important parts):

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Qarl wrote:

• Generic modifiers to damage dealt will now apply to damage over time that your character causes. Damage over time is not spell damage, nor attack damage, so modifiers specific to those types will not apply. Non-specific "increased damage" modifiers will apply to damage over time.

Minion Damage is not a method of dealing damage. It's a generic modifier ("increased damage" with no specifiers), that is applied to your minions.
A spell is something you're doing that might cause damage. A minion is a separate entity in the world that does it's own things that might do damage (and may or may not be spells). These two are NOT interchangeable in the way you're trying to do - the entity which has dealt the damage (minion/you) and the way that entity dealt the damage (any/spell) are different concepts.

The confusion, I believe, stems from the fact that in English, if you cast a spell on me and it hurts me, it would be equally valid for me to say "You damaged me with that spell" or "your spell damaged me" - and thus in general conversation we commonly talk about damage being dealt "by" the spell when in fact it was dealt by the player, using the spell as a means to deal it.
I'm going to try to be more exact in this post, but this inherent ambiguity means that any other quote you read on the forums may well be referring to the spell as "dealing" damage, simply because that's more understandable than referring to the player dealing damage with the spell, and in the original context, there was little or no room for misunderstnading. Taking those quotes out of context and comparing them to others which refer to the entity which actually dealt the damage and not how they did it could lead to confusion, as I believe it has here.

So to go back to your edit of my quote:

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acme_myst wrote:

The ignite is fire damage over time dealt by a minion(spell). That's a subset of damage dealt by a minion(spell), and thus minion(spell) damage applies.

This is wrong. Here's the correct expansion - the underlined bits are, like in your example, what you'd swap in for the spell case; the bold bits were added to give them something to swap for, as the relevant parts were irrelevant in the context of the original quote and thus skipped.

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Mark_GGG wrote:

The ignite is fire damage over time(which is not spell damage) dealt by a minion. That's (not) a subset of damage dealt in any way(spell) by a minion, and thus minion non-specific/generic(spell) damage applies (does not apply).

In case that was confusing, I've provided the two versions separately in the spoiler

The equivalent of "minion damage" is not "spell damage", it's "player damage" or "your damage" which in game is just referred to as "damage" - which is not specific to spells (or attacks or any of the other categories) so applies to any damage you deal, including damage over time, just like "minion damage" is a modifier to all damage the minion deals with it's spells/attacks/etc.
If you gave a minion "increased spell damage" (which can be done easily with Necromantic aegis). It would only apply to spell damage the minion deals, and thus not to any ignites caused by it, since they're damage over time.

Spell damage is a modifier to one way you can deal damage, and explicitly not to other ways you can deal damage. Minion damage is not - it's a modifier to any damage, regardless of how it's dealt, dealt by a separate entity (the minion).

For the record, In my experience, if Vipermagi says something different to the wiki, The wiki is more likely to be in error. Vipermagi is certainly not infallible, but tends to be more up to date and more able to understand in new knowledge from patch notes in light of previously known knowledge than the wiki.