I also don't know what the weave is like in 5th.
I do know that in both 2nd & 3rd that the weave is both the interface and Mystra herself. In the primordial battle, when Shar was about to kill Selune, Selune made one last ditch effort to repel her sister. Selune transformed her body into a blast of raw holy radiance that tore through Shar. The chaotic, luminous substance that passed through Shar's body took portions of Shar's darkness with it and that god-stuff formed the goddess Mystryl. Mystryl joined with Selune to push Shar back. That pattern of light and dark, visible as the stars in the sky, was the weave and Mystryl herself.
After Karsus killed Mystryl, the newly formed Mystra decided that the weave was too open and too chaotic and so she began to untangle the chaotic skein and to order the threads on a divine loom. Wizards who invented new spells aided this effort, every new spell brought order to the chaos of the weave.
Meanwhile, Shar nursed old wounds and traced the patterns that Selune's blast left behind until she we was able to create the mirror opposite of weave from the mold of her own scars. The shadow weave is the anti weave, a cancer on the weave. Where weave and shadow weave interact shadow consumes luminescence. Wizards who cast too many shadow weave spells find themselves unable to connect with Mystra's weave. It's never been clear if the shadow weave is the mirror of Mystryls (infinitely more powerful) weave or if the shadow weave adjusts itself as the weave is ordered. The fact that the Shadowvar were able to keep their city flying suggests the former (because that's a 10th level spell) and the fact that the shadow weave died when Cyric killed Mystra says the later.
BG3 is the first mention I've heard of the shadow weave -- I thought it was still gone.
TL;DR -- Gale almost certainly has an portion of Shar's shadow weave and not Mystryl's weave because it is acting like the shadow weave and not the Netherese weave.
WotC released Nethril: Empire of Magic into the public domain loooong ago (before they decided that it was mistake to make their old stuff public). You can still find it on the web. Much of the Mystryl - Mystra story is told there.
I'm really happy the shadow weave is back -- one my favorite parts of the lore.
