That's roughly what was said, but it was contradicted by treating psionics like magic and allowing them to be dispelled and rendered useless by anti-magic. Personal psionic power wouldn't be affected by those things.
In 1e and 2e psionic was considered nonmagic.
The 3e Forgotten Realms setting invented the concept of a "Weave".
But the 3e default was, magic and psionic were "transparent", meaning they werent the same thing, but psionic interacted normally as-if magic. When antimagic was able to suppress psionic powers, there needed to be an explanation. Psionic too has a kind of weave, even if it is unrelated to the multiversal Weave. Hence, each psionic has ones own personal weave, an aura around the psionic creature. Antimagic suppresses this aura. (Or as an optional variant, antimagic is unable to suppress a personal weave.)
In 4e, psionic is understood plainly as magic. But psionic had a separate and independent "power source" of magic. In Forgotten Realms, the tradition of a personal psionic weave continued. The personal aura is the psionic power source.
In 5e, Forgotten Realms is the default setting. The multiversal Weave exists, whether by this name or by any other name. However, psionic is officially independent of this Weave, and its source is ones own "mind". A mind emanates its own magic effects.
Similarly, the "ki" is its own source of "magic", independent of the Weave, where antimagic cannot suppress it.
Curiously, in 5e, psionic can do things that are "spells", and psionic can do things that arent, such as Psi Warrior features. Antimagic can suppress the "spells", but it cant suppress the nonspell magic. The psionic magic itself is independent of Weave, and persists despite a void in the Weave.