The Ashbringer does have runes inscribed on it in both its purified and corrupt form, and the runes appear different in both (with the former having a neat, blocky script and the latter having a loose, cursive-like script). The runes on the corrupted form could easily be capable of channeling the magic on which death knights rely, and Mograine was very quick to replace his weapon once he relinquished it.
Darion does honestly kind of fall between the two definitions. His death was a suicide by the corrupted blade, and it’s kind of unclear if the blade itself brought him into undeath or if Kel’thuzad did but the implication seemed to be it was the blade. So his undeath is very much tied to that particular sword (a very unique sword, at that; likely crafted from a Naaru fragment or related artifact), putting him in a little category above “typical” undead as it is.
But the definitions are pretty loose and just kind of a simple guideline. He’s a death knight. He died, came back, put on spooky armor, and used a goofy sword that is in fact a runeblade! Just not a Scourge runeblade, but something very special and easily swayed by outside influences due to its dual, possibly-Naaru-related nature.
Bluh bluh, incoming stupid paladin lore blurbs.
The Ashbringer is indeed a runeblade! King Magni Bronzebeard himself forged it, using dwarven runes. That is a thing they do, making crazy weapons like Ironfoe. The end result was a blade of legendary hardness and sharpness.
However, because of the big dumb wind chime heart they used to make the thing holy, it’s incredibly susceptible to corruption. When Renault killed Big Papa Mograine with it, it was corrupted and for all intents and purposes became a Vampiric Runeblade, which is what death knights use. Soul-eating, life draining weapon infused with incredible unholy power.
So when Darion stabbed himself with it, it ate his soul. Just like any other vampiric runeblade. And that pretty much sets the stage for becoming a death knight, hard to do anything else once you’re stuck inside of a big dumb weapon.
Funny enough, he more or less literally threw his soul at Tirion when he chucked the Ashbringer at him. Instead of wiping him out, it probably popped his soul out like a cork from a wine bottle since the Ashbringer stopped being a Vampiric Runeblade and just went back to being a nearly indestructible and insanely holy masterpiece. Lucky him.
TL;DR! The Ashbringer starts out as a Dwarven Runeblade, becomes a Vampiric Runeblade when it kills Alexandros, and returns to normal when Tirion lays his clammy old hands on it after it snacks on the entire Mograine family tree.
Here is a better explanation of Ashbringer. THANKS NERD.