That is an interesting philosophical question that has not been conclusively answered in canon. You are unlikely to get a consistent answer from the forum.
I interpret it like this:
The Gift is a particularly strong magical ability (or potential) that exists within people. It is sort of a meta-ability in that the Gift is not the ability to do anything specific, but rather the ability to easily learn a wide variety of different magical abilities.
As you say unGifted people might learn a few magical powers here and there but each of these requires an extraordinary piece of work on the part of the person learning the ability, such as extensive training or an arduous task (mechanically simulated with an intiation). In a way I think many ungifted people who can use magical abilities are either using the magical ability of some other entitity such as ungifted initiates in cults who I assume are actually wielding their patrons powers on behalf of the patron OR they people who are adept at utilizing natural magic. This includes people like skinchangers who use natural sympathetic magic to their advantage or dowsers who are adept at reading magical energies.
It is reasonable to assume that from the perspective of the characters in the game, at least from the perspective of hermetic magi, that the gift exists on a spectrum, some people have strong gifts and some have weak gifts. The mechanics differ but that may just as well be a necessary abstraction because the alternative would be to have a "Gift score" for each character that measures the strength of their gift. In that context the virtue "The Gift" should be considered a statement of "your character has a Gift sufficiently strong to be trained in hermetic magic" rather than a statement that the Gift is essentially binary. Many hedge traditions dont fit neatly into hermetic categories and so it is often hard to determine whether the Gift is necessary to learn the powers of specific traditions, other than to say "If the hedge tradition has strong, varied and/or flexible magical powers they need the Gift", which is IMO a fine solution. Exactly where you draw the line is unfortunately a matter of adjudication that you will have to decide on yourself (along with your troupe of course).
However as stated above, perspectives on this may vary widely, and I am sure that you can assume any number of equally legitimate viewpoints of the underlying metaphysics (metamagics?) of the Ars Magica universe.