Maybe. I would need to review the specific wording here and there in the official texts.
At least the 5e designers freely describe every form of Psionic, whether spell or nonspell, as "magic", and refer to Dragons as inherently "magical" despite the antimagic not applying.
With regard to antimagic, including the spell Antimagic Field, I would delete the concept of a "Weave" from the D&D game. Instead, the antimagic suppresses any "new" magical phenomena. But any "old" magical phenomenon that lacks a magical duration continues normally. Suppose a +2 Flametongue Sword. The +2 is persistent, thus the antimagic cant suppress this, but the antimagic an suppress evoking the flames because that would be a new phenomenon. The presence or absence of a hypothetical Weave is irrelevant.