I've been researching the magic lore of DnD for a possible DMsGuild side project to be published perhaps in between The Trial of Asmodeus and Realmspace Traveler's Guide.
The Weave is often mistaken as the "origin" of magic, when it is actually a tool for channeling magical energy and assisting calculations when programming the algorithms of spells. Mystra's Weave, the one featured in the Forgotten Realms setting, would be only one of the Weaves that can be accessed throughout the multiverse, generally limited to Realmspace and overlapped regions of other planes.
Magical energy flows almost everywhere throughout the multiverse, independent of Weaves (or rather Weaves depend on the presence of magical energy to exist instead of the other way around), more heavily concentrated in some areas such as leyline hubs.
Psionics is a method of spell casting by forming a personal Weave within one's mind.
Epic (seed) magic is Weave magic, but with extra steps compared to standard casting/rituals (the seed elements), which is why "Mystra's Ban" against 10th level or higher spells can be relaxed with Epic magic (it's easier to regulate due to the extra steps involved).
Elven High Magic is non-Weave magic that is complex and can be interfered with by Corellon Larethian. My personal hypothesis is that Elven High Magic is a middle ground between Psionics and standard casting, in which the casters form a temporary environmental Weave, which is potentially dangerous for the casters due to the difficulty of stability maintenance over energy flow, as well as the number of variables that must be calculated in the algorithms of the spells cast.
Sources: Ed Greenwood, Player's Guide to Faerûn, Epic Level Handbook, 5e Player's Handbook, FR novels including the Avatar series and Spellstorm