I've been in topics like this before.
The short answer to how I interpret Link - read my fan fiction.
The longer answer... (going by the actual canon games, not the cartoon), I see the various Links as having similiarites (a "base soul" ) and some differences. I see them all as brave, first of all (this is the core trait), fairly humble (word of Miyamoto on that, he's said in an interview somewhere that he envisioned Link as a "brave but humble young man). I see them as essentially independant souls, valuing personal freedom highly. They also care very deeply for friends/relations and want to do the right thing, always. There's always a hint of mischief in them - I see them as maybe a bit more Chaotic Good or Neutral Good than Lawful Good. They like shiny weapons and exploding things way too much to not have a little "chaos" in them. My Links tend to think a lot about what they must do to save the land - in other words, even though they must kill things, any given Link is a "soldier" not a "homicial maniac." Above all, Link is a noble soul.
Some of the differences between them as seen in my fan works:
Original LoZ/AoL Link: The traveler who gets caught up in the troubles of a land not his own (Hyrule). Takes up the sword because he knows that things "just aren't right" and cannot stand to see good people suffer. Good at wilderness-survival.
ALTTP/LA: I haven't written him very much, actually - mostly a small spot in a larger fan fiction. Kind of a confused kid caught up in things he doesn't have much understanding of, but brave, tough. (as I wrote him, a bit guilt-ridden over the fate of a certain island).
Ocarina of Time and Majora's Mask: Young Link is very serious for his age. He strikes me as a rather serious child, but that doesn't mean that he doesn't enjoy play. He's very playful, he's just more responsible and even grim-minded (especially in Termina/after Termina) than the average kid of 10-11-12. I also like the quirky idea that he does not know how to read. This was taken from a friend of mine, who noticed that the Kokiri (and Link) seem to use fairies to read all the signs.
Adult Link - I do think he matured a little in spirit as well as in body during the magical seven-year's sleep, because there was probably some necessity for it, but mostly, he is a young man with the memories and essentially the soul of a child. It probably enables him to retain the "pure heart" he needs as the Hero of Time. He's rather playful and interested in learning. He has a very ardent desire to "protect." Will of steel.
The both of them, in one- Adult Link has the memories/heart of a child, while Young Link - post being sent back in time by Zelda, has something of an adult heart (from having been one for a while) in a child's body. This makes him a bit of a confused, dualistic being. He takes it in stride, though. I also tend to like the idea of him getting a bit twitchy on nights with full moons after Termina, and suffering nightmares leftover from his experiences. Also, in my fanfictions involving his death, he has an unfortunate tendency to linger as a spirit because he wants to continue to protect people even when he lacks any real power to do so anymore.
Wind Waker Link - A brave, ENERGETIC little boy who loves what family he has left more than anything in the world and will beat you sensless if you lay a hand of harm on his little sister. When I played that game, the kid just struck me as a little ball of energy. Probably quite optomistic.
Twilight Princess Link - Responsible, hardworking and tough. A very kind soul, but also with a touch of grit. I seem to write him (in my fanfictions) as being able to come to terms with things fairly easily - he's had a lot of hardship in life (including a running thing in my fanfictions where he survived a plauge that killed his parents). Values freedom highly. Also very bright (I think all the Links are very bright) and unlike the Hero of Time, an avid reader from a young age (he has many books in his house). Well-versed in old legends, including the ones about the Hero of Time. Tends to be a bit of an artist, likes to sketch and draw. In fanfics of mine where he dies, he tends to accept his death easily, often "moving on" easily, or lingering to help out friends, but being at peace about everything.