Heidegger was reacting to Being as objective and theoretic; that is the being of science - atoms, matter and forces. Cassirer for example his most prominent opponent saw science as the culmination of philosophy; Heidegger wanted to re-orient philosophy to being here in the world - throwness and this gives a different aspect onto notions of time, space and ethics.
Authentic derives from the Latin authentes, one acting on ones own authority; and from the greek auto, the 'self' and hentes, doer and being.
Primarily it is only Dasein that can be authentic or not.
We are not talking about the authenticity of a forged or not painting by Van Gogh.
Daseins possibility of being authetic or inauthentic is rooted in the fact that Dasein is always mine...Since it is mine I can lose it or grasp it, for the fact that it is mine does not mean that it is 'properly' ones own.
To have a hammer, but not to know how to use it, is in a sense not to have a hammer. Thus to have Dasein is not to grasp Dasein 'properly'.
Dasein doesn't lose itself as it might lose an umbrella. It does so by 'falling concern'. It falls into and is absorbed by the 'world', so that it forgets itself as an autonomous entity and interprets itself in terms of its current preoccupations...Despite all this, Dasein is still concerned about itself. If it ceased to itself to matter at all, it would cease to 'Care' (Sorge) and would lose all concern (Besorgen) for anything.
The 'world' I suppose can be an inner as well as an exterior world.
Inauthenticity is only a 'modification', not the extinction of care. Dasein is never irretreivably lost in inauthenticity. If it were, it would no longer be 'inauthentic' as it would no longer be Dasein.
Dasein is defined as that Being that cares about being. Thus to no longer care is no longer to be Dasein.
"Is the idea that we can establish the meaning of our death"?
I don't think that this is Heideggers concern with authenticity. At least its not mentioned by Innes in the entry for authenticity; its remarked on in his entry on Being; and then not even as a separate entry and not hugely relevant:
Heideggers preoccupation with death does not survive the BT period...and did not advocate obsession with death...It does not affirm nihilism or the senselesness of being.
But then, so what?
"Not everyone need perform Being-toward-Death and assume the Self of Dasein in this authenticity; this performance is neccessary only in connection with the task of laying the ground for the question of being; a task which is of course not confined to philosophy. The performance of Being-toward-Death is a duty only for the thinkers of the other beginning, but every essential man among the future creators can know of it". (LXV 285).
Innes comment on this is that running ahead to death primarily secures the integrity and the self-constancy of the individual and not to questions of Being; or rather indirectly so; for those in which it is a duty ('thinkers of the other beginning') Being reveals itself - but it is unclear to Innes how this is so or happens; it is only clear that detachment from 'everydayness' (an aspect of Authenticity) is a necessity.
This counts almost everybody out. Authenticity is more about the meaning of life and not death; in living as though it was your first day and not your last. Innocence and its fullness when one is alone and not alone in the world as in Shelleys Spirit of Solitude; the 'green world' of Shakantula. In the everyday trite phrase self-fulfilment or in the Greek eudaimonia.
Or is it is more of a +ve reflection on the authentic person's character, that they encounter their own death and not flee from that anxiety?
In part yes, in part no. Innes says:
Inauthenticity is not a moral or a theological notion (XXI 232;LXV 302)
Thus its converse - authenticity - one supposes is also neither a moral or a theological notion. Thus not a reflection on an 'authentic persons character'. For the individual, encountering their own death, 'running ahead to death' in Innes terminology 'secures their own integrity' as a person (integrity is not being used here in its moral sense) or as you say 'not fleeing from that anxiety'. Its referenced in Blakes poem The Book of Thel.
What's the point of authentic being towards death?
The main point is as you say to secure the integrity of the person - that is for most; and I'm speculating here that this would be confirmed by a close examination of the anthropological literature on rites-of-passage; for a few ('thinkers of the other beginning') its related to the discovery of Being; in Badious terminology Being as an Event. This is neccesarily obscure; as its encounter can only be for those who have a 'duty' to meet it: In the Christian tradition, this is exemplified by Abraham/Isaac & Christ on the Cross; in the Islamic tradition of the Miraj which influenced Dantes Divine Comedy; in the Western philosophical tradition by Parmenides, Pythagoras, Plato, & Plotinus.
(Quotes from Michael Innes Heideggers Dictionary)