A distinction exists
In the rules on damage resistance, the following example is given:
Multiple instances of resistance or vulnerability that affect the same damage type count as only one instance. For example, if a creature has resistance to fire damage as well as resistance to all nonmagical damage, the damage of a nonmagical fire is reduced by half against the creature, not reduced by three-quarters.
First of all, this example gives evidence that "resistance to all nonmagical damage" is a valid type of damage resistance, and that the phrase "nonmagical damage" is itself a distinct category of damage (presumably defined as "damage that is not magical"). Additionally, because the example is being used to demonstrate that multiple applicable resistances do not stack, we can conclude that "resistance to all nonmagical damage" would grant resistance to damage caused by a nonmagical fire. If it granted resistance to damage caused by a magical fire as well, then the example wouldn't need to specify that the fire in question was nonmagical, thus I assert that "damage caused by a nonmagical fire" can be called "nonmagical fire damage" and "damage caused by a magical fire" can be called "magical fire damage".
Is "magical" a damage type?
"Magical" is not a damage type. The rules on damage types are short, in fact they mostly say "Damage types have no rules of their own", but they include a list of all the damage types and "magical" is not one of them. If you look through all the books you will not find an effect which simply "deals magical damage". It never exists in isolation because it is not a damage type.
What is "magical damage" if it is not a damage type?
"Magical" is a modifier that can be applied to damage, just like it can be applied to almost anything else. Fire created by magic is "magical flames". Darkness created by magic is "magical darkness". In the same way, damage caused by magic is "magical damage" even though "magical" is not a damage type and the damage also has a type that isn't "magical".
As an example lets take a +1 longsword. Nonmagical longswords deal slashing damage. Magical longswords also deal slashing damage. However, Nonmagical longswords deal nonmagical slashing damage while magical longswords deal magical slashing damage. Thus, a +1 longsword deals magical damage.
Maybe the example in the Damage Resistance rules is a mistake?
Although I can find no references directly to "magical damage", there are several cases of resistance to "nonmagical damage" or a subset of it, and which do not use the "damage from nonmagical attacks" phrasing. From things listed in the Basic rules, the spell gaseous form grants "resistance to nonmagical damage", the spell stoneskin grants "resistance to nonmagical bludgeoning, piercing, and slashing damage", and the magic item armor of invulnerability grants "resistance to nonmagical damage". Looking outside the Basic Rules, Demon Lords from Mordenkainen's Tome of Foes, such as the Demogorgon on page 144, have Damage Immunity to "bludgeoning, piercing, and slashing that is nonmagical." I find it unlikely that all of these are mistakes.
Maybe it means something else?
The most authoritative source available for how to determine whether an effect is magical comes from the question "Is the breath weapon of a dragon magical?" answered in the Sage Advice Compendium. There is a lot of information there, but the most relevant part is this sentence:
The breath weapon of a typical
dragon isn’t considered magical, so antimagic field won’t
help you but armor of invulnerability will.
Thus armor of invulnerability is used as the canonical example for "protecting you from damage caused by a nonmagical effect". I'm not sure how much clearer it could be.
The phrase "nonmagical damage" is officially used in multiple places to mean "damage caused by an effect/feature/item that is not magical", using the typical definition of "magical". To complement that, the phrase "magical damage" would sensibly mean "damage caused by an effect/feature/item that is magical", though I haven't found that phrase used anywhere. Thus it is sensible to talk about "resistance to nonmagical damage", and "ways to cause nonmagical damage".