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If an ability would produce one or more mana of an undefined type, it produces no mana instead.

Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.

To “tap [a permanent] for mana” is to activate a mana ability of that permanent that includes the {T} symbol in its activation cost. See rule 605, “Mana Abilities.”

An ability that triggers whenever a permanent “is tapped for mana” or is tapped for mana of a specified type triggers whenever such a mana ability resolves and produces mana or the specified type of mana.

A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.

If an object’s enters-the-battlefield triggered ability or replacement effect refers to X, and the spell that became that object as it resolved had a value of X chosen for any of its costs, the value of X for that ability is the same as the value of X for that spell, although the value of X for that permanent is 0. This is an exception to rule 107.3i.

The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The untap symbol is {Q}. The untap symbol in an activation cost means “Untap this permanent.” A permanent that’s already untapped can’t be untapped again to pay the cost. A creature’s activated ability with the untap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

A planeswalker is dealt excess damage if it's dealt damage greater than its current loyalty.

It doesn't matter whether a Giant, Wizard, or spell you control deals the excess damage, only that excess damage was dealt and that one of those three things dealt damage to the creature or planeswalker at some point during the turn. For example, if a 4/4 creature an opponent controls is dealt 2 damage by a spell you control and later that turn is dealt 3 damage by a spell another player controls, Aegar's ability will trigger.

If a permanent is both a creature and a planeswalker, the minimum amount of damage to be considered lethal damage is used to determine if excess damage has been dealt. For example, if a 5/5 creature that's also a planeswalker with three loyalty counters on it is dealt 4 damage, it's been dealt 1 excess damage and Aegar's ability may trigger.

A creature has been dealt excess damage if one or more sources deal more damage to it than the minimum amount of damage required to be lethal damage. In most cases, this means damage greater than its toughness, but consider the damage already dealt to it that turn.

Even 1 damage dealt to a creature from a source with deathtouch is considered lethal damage, so any amount greater than that will cause excess damage to be dealt, even if the total amount of damage isn't greater than the creature's toughness. Note that a source of damage having deathtouch has no effect on damage dealt to planeswalkers.

Aegar's ability won't trigger when a battle is dealt excess damage. It's not his fault. They didn't exist yet.