Actually aurora, it's more nuanced than that. Notice how in your quote, it says 'understanding', not 'sense' or 'awareness'. It's a key distinction.
The Weave is the medium through which most mortals access magic; it's not essential for magic (
Raw magic exists in all things, but cannot be safely tapped in that way - for a mortal to attempt to do so would in most cases scour them to cinders as well as being completely beyond control or direction), but it's as near as being so, more or less. The exception here is Sorcerers who have the potential to draw on raw magic directly without using the weave; it's dangerous and unpredictable, but for those with that innate magic within them, it's at least possible - this is why sorcerers exist in revered and reviled positions on Abeir, where these is no weave; other spellcasters on Aebir - even divine casters - must go to extraordinary lengths to gain access to workable magic, such as making pacts with intermediaries or by the extremely pains-taking crafting of magic items that can serve as specific magical intermediaries for specific purposes.
On Toril, where the weave exists, it still isn't something that everyone can 'sense' or 'feel' - that's not a faculty that most mortal creatures have, at all. Despite common house-ruling by DMs, you cannot pick up a magic item and 'feel' or 'sense' that it is enchanted; you simply can't. You can observe signs that suggest it is (an inexplicable glow, an unnatural resilience or impossible construction, a defiance of rust, age or wear amongst other things that show their age, and so on), but you cannot just 'feel' that it is magical.
Magic users, with the exception of sorcerers,
Cannot simply sense magic or feel the weave. There are spells that let you do so, but without them, you can't.
For casters who use an intermediary (whether that be a divine boss, like a cleric, or a business relationship like a warlock), they can feel the reverberations of magic working through them, and either by force for will or intuitive understanding of what to do and how to work with their intermediary, can direct and control these effects, but what they feel is the second-hand pressure - the work of their intermediary channelling and focusing magic through them, not the magic itself.
For wizards, it's a step further removed: A wizard does not have an intermediary to channel magic for them; they do it themselves, and they do it blind, trusting upon their own understanding and intellectual precision of word, motion and material, and the academic knowledge that certain combinations thereof cause folds and wrinkles in the weave in certain ways; the level of academia required to understand and process this level of detail is very near to beyond our comprehension, and goes above and beyond even most modern scientific pursuits.
But, they must do this, because they cannot innately 'feel' or 'sense' the weave at all: they have no innate magical ability, and that's the amazing thing about wizards - anyone with the brain power for it can become a wizard, because there is no natural innate ability required. Few do, because few have the demanding mental faculties required to go very far in the field.
Think of it this way: To create a water fountain, you need to build the fountain itself, and you need to construct the pipes that lead from the reservoir to the base of the fountain. You know that the reservoir is there behind the wall, but you can't see or feel it. you know that if you build a certain set of pipes in a particular way, however, it should connect to it, and if you build the fountain a certain way, the effect will look the way you want; this is academia that you've studied. So you build the pipes and the fountain, and then you put the last piece in place and twist the handle, laying your hands on the pipes as you do. You do not feel the water itself; your hands don't get wet, and you can't see it or feel it rushing through your fingers directly... what you do feel is the reverberation and rumble of that water rushing through the pipes you built, and then you see the fountain burst to life, just as you calculated it should. That is wizarding.
If you're a sorcerer, on the other hand... you don't really know much about water reservoirs or pipes or fountains... but you can feel a large amount of water just behind that wall, and when you reach out to it, you can feel it shift in response to your call; you can pull it from that reservoir, trough the pattern you can feel it needs to flow though to make the beautiful fountain display that you want, guiding it through the extended senses you have and forcing it to conform when it tries to escape and splash everywhere, by your own force of presence - which it responds to. The wizard/plumber wants to know how you did this, but you can't really explain it to them in a way that makes any kind of sense to them - they want to know by what mean,s by what tool, by what faculty were you able to reach out and manipulate the water, but you cannot explain it any more than saying that you just do it; it is innate to you, and it's not something that can be 'learned'; the wizard cannot 'decide' to become a sorcerer - they simply are not one. You can describe to the wizard what the power feels like as it flows through and around you, and perhaps even see the longing in their eyes as you do so, but you cannot help them or teach them, because you are describing flexing muscles that they simply do not have.
Sorcerers do, generally, utilise the weave in a similar way to others - one because that's mechanically comparable for a game system, but also because it's easier and offers the path of least resistance and smoothest communication for creating effects... but by the lore, external to playing the actual game, they don't *strictly* need to in the same way that other casters do.
