The SAC draws a distinction between "magic" and "supernatural." Some abilities are properties of a supernatural nature but aren't explicitly magical. Per the SAC, only abilities that are explicitly magical are suppressed by the field.
This is probably the best counterpoint to my arguments that "of course teleports are magical". At the end of day, we already accept that some forms of magic (aka what keeps undead going) just keeps on trucking in an anti-magic field. So once you accept that, its really not a question of "is teleporting magical"....because of course it is, the real question is "is it the kind of magic actually blocked by a anti-magic field?"
That’s why Sage Advice differentiates between supernatural abilities that are an inherent part of a creature’s nature and Magic. I can accept that there’s a distinction between the two, but it does seem super weird that those four abilities Vecna has fall into the former category, especially given that his lore is that he was a human who became a powerful undead through the use of evil magic.
Still pretty subtle. I think it means that your casting is, relative to normal casting, very subtle. It doesn't mean that it's so subtle that a guy with hundreds or thousands of years of spellcasting can't tell what you're doing. Spellcasting takes effort.
It conjures "spectral entities" so is similar to Spirit Guardians in many ways, just a one-shot.
The other two abilities are pretty must Misty Steps when he teleports: "you teleport up to 30 feet to an unoccupied space that you can see."
Anyway, lot of stuff in 5E omits individual words, trusting people to understand the intent.
Antimagic doesn't stop something that IS magic (such as a dragon or undead) from functioning, but his features are active magic, as @FitzTheRuke suggests.
1. Malek uses Cunning Action to Dash to Vecna, dropping his bow and drawing his rapier.
2. Malek attacks with advantage, rolling 3, 11 + 11 = 22, hitting Vecna's AC 18. However, due to the Antimagic Field, Vecna's immunity to non-magical piercing damage suffers no injury.
3. Vecna screams in apparent pain, and then seeing the smile on Malek's face, laughs manically, "Fool! Inside this field you cannot harm ME!"
Egar calls to Malek, "We need acid or fire or something! Anyone have torches or Alchemist's Fire?"