The Gift is discussed on p 75-77 et al, but you're right, it really doesn't talk about what The Gift changes in the Gifted themselves, on a more personal basis.
Part of that I alluded to above (What difference does the Gift make? - #7 by Cuchulainshound), how it affects Gifted children with uncontrolled (and, to the ignorant, unexplained) outbursts of spontaneous magic. That, alone, will "mean" something to the developing mage.
But I suppose the essence of what The Gift means is that it opens a door to part of the world unlike anything any non-Gifted can appreciate. It is (almost literally) as if you were the only person who could see "cause and effect" in a world of superstition and ignorance. And not only can you see and understand this otherwise invisible, unknown and unappreciated force - you can control it.
What that elite insight and power then does to shape the psyche of the mage varies widely, as one might expect - some go the "With great power comes great responsibility" route, others go more ego-maniacal, some sociopathic (to one degree or another), and still others seem largely unchanged from what one might term "normal". It's a combination of Nature and Nurture - each individual varies, but also much depends on their early, pre-apprentice experiences as well as the 15 years of the apprenticeship itself.
(Is that more the direction you're asking?)
Yes and no...
Not all founders had "been instructed", and not all were part of "traditions". Think of what I spoke of just before, of a Gifted child growing up with... all that comes with it. Seeing inexplicable things occur around them, at times being able to control those phenomena. Now remove any Hermetic training - they're on their own to "figure it out". Perhaps they survive Twilight and gaining some insight - but they have no terms to define it, no words to describe it, no model to build upon nor framework to hang it on. Anything and everything was new and inexplicable, even if it was real and tangible - if only to them. And because of this, that child, that growing mage/sorcerer/hedge wizard, would create their own tools and skills and "mechanics" to make magic work for them as best they could, as they needed and understood it, and with or despite whatever Virtues and Flaws and quirks they took for granted within themselves. And if that hedge wizard went on to establish a "tradition", their pupils would grow up with magic accepting their teacher's model as gospel, and accepting that any other wizards were utterly and irreconcilably alien and incomprehensible.
The Founders, Bonisagus included, came from that type of background. Some were indeed not first generation, but they only knew what their teacher had taught them, and that teacher only knew what s/he knew. What Bonisagus did was two-fold - he translated all those different traditions and understandings into a single unified framework of "Hermetic Magic", and, in so doing, he gave them all, and every Hermetic teacher and apprentice to follow, a comprehensible model and vocabulary that could be shared between them all.
And in a way, that's exactly what had to happen to achieve all that.
For every understanding that got shoehorned into the Hermetic model, some parts of that "tradition" (be it first-gen or centuries old) had to be sacrificed to fit. It might have worked fine before, maybe even in some ways "better", but if "the Hermetic Model" is to be successful in its goals of universality, non-Hermetic aspects are filtered out, for better or worse.
Which is why many non-Hermetic magi, and almost all non-Hermetic traditions, are, understandably, not real keen on The Order.