Okay, so given sorcery is privilege and institutional power as its metaphorical role in this re-work, then Lineage represents inherited wealth and "getting a good start in life". It's metaphorically the kind of cultural, economic and social capital you get by not being born poor.
(incidentally, what Salina does becomes metaphorically guaranteeing everyone a university place)
So a Lineage isn't strictly "things passed down to you by your teacher". It's more, mmm, the weight of this capital. So when you get Lineage from your Gold Faction mentor, you're not just getting what she taught you. You're getting the weight of the oaths that she's sworn with gods, and that her teachers swore with gods, and all those spirit negotiations and spells that her predecessors invented and countless things like that. You're getting things like "Oh, I recognise how this spell works because the school of sorcery I learned invented it". It's a mess of soft benefits and social expectations; it's metaphorically "having the right accent" and "knowing how to interview well".
The Gold Faction mentor therefore gives you a super-comprehensive Lineage that comes from the Silurian traditions of how Sidereals learn, has multiple people who were taught by Silur herself, and so yes, when you cast an Adamant Circle spell you do it in the Department 131 Silurian style and it's very characteristic, drawing on the weight of the schools of heaven. By contrast, the gods are a far less classy lineage and you might be more thematically restricted as to what you can use it with - but that's probably represented by it being a less prestigious lineage (ie, lower-rated) and so you can't use it for high rated spells. It's perfectly functional for TCS, though.
(Imagine sorcerers getting snobby about what university they went to. "Oh, that's nice, you went to a former polytechnic. I studied PPE at Oxford, don't you know?" That is exactly what Lineage is like in-setting)
Unfortunately, no, you can't get Lineage just from notes. Those are "components" you can use for justification for making or altering spells. On the other hand, you could get it if there was a spirit bound in old ruins oathsworn by Silur to teach any Solar who comes looking for wisdom. I mean, mmm, okay, maybe you could get it from, like, a sorcerer's personal grimoire or whatever, but by that point it's not just notes. It's more that you've just found the Book of Darkness (and its in-built Familiars), which is also an artefact in its own right. It's got to be a comprehensive teaching aid to get the benefits of Lineage.
However, yes, you can upgrade your Lineage by finding more teachers. For example, if your Solar is first taught by a rustic god of scholars who long ago swore to help the Solars and has been teaching them whenever they show up (getting Lineage 2), then if they later become an Infernalist and study under one of the demon princes, then their Lineage goes shooting way up. Learning from demon princes is one of the easiest ways to get a high Lineage in modern Creation, because the lines of learning which pass through Third Circles are - ha ha - unquestionable.
It's one of the things they use to lure sorcerers into infernalism.
Or just localised?
