...So -- remembering we are talking about the fiction, not the rules -- what is a spell in D&D, and what isn't?
Just to mention it: I'm not currently adopting the Mordy's changes and am sticking with the designs to the monsters/NPCs that feature more spells and less abilities. If I do, I will likely generally be 'homebrewing' to add more spells to those creatures and then give some of their abilities a 'spell' tag so that those abilities can be counterspelled, etc...
I know what they intended to do with these new designs, but I consider them major missteps. They messed with the mechanics of the game in a way that does not work with existing rules. It smacks of an 'end of edition' mass playtest, so we've begun to enter the phase where I am less likely to purchase new materials. It disappoints me as I think we could run 5E for another 5 to 10 years ... it is a good edition.
I wish they'd just kept the monster sheets as they were but bolded the options they'd recommend using first. Eliminating monster options on such a massive scale has been a huge mistake every time they've done it - and they've done it over and over. By 7E we're going to end up with only one monster with one attack in the monster manual at this rate.
That being said, historically, magical monster abilities in my campaign come from Supernatural Magics as a general rule.
In my setting there are 5 types of magic: Arcane, Divine, Nature, Psionic and Supernatural. Arcane, Divine and Nature magics utilize the Magical Weave that extends from the Positive Energy Plane to the Negative Energy Plane (each in a different way). Psionic magic originates within the creater of the magics.
Supernatural magic, however, is drawn from the universe around you, but it does not come from the weave. It covers things like undead, curses, chemistry, and monster abilities not identified as a spell. Historically, most of these are not cast are manipulated into existence. They tend to just happen. There are exceptions, but that is the general rule.