The spells produce magical effects, like a wall of fire that needs no fuel. My ranger could do no such thing.That's circular, and means nothing. It's magic because it's an ability being used by a magic using character, but the character is a magic using character because they use abilities which are magic.
So, it's magic because the character using the ability says "I am casting a spell". So if a barbarian says "I am casting a spell" and uses Reckless Assault, Reckless Assault is a magical ability, and the barbarian is a magic user. Channel Divinity is not a magical ability, unless the cleric says "I am casting a spell" when they use it.
Magic-using characters are outlined as such in the rules, they are described as using magic. It's right there in the class description.
The spells the wizard had in the party are called out as such: magical spells. Reckless assault is not called out as a magical spell. When I say the player says "I am casting a spell," it's because it is outlined as such on his sheet and in the rules.
This is not complicated.