I've just thought of another way of thinking about this, which may (or, then again, may not) be of help. (As someone who only started to play almost exactly a year ago, I still welcome the chance to be made to think more deeply about things like this, even though it must be obvious to the experts
). An "eye", often defined loosely in terms of a point (or set of points) completely surrounded by one's own stones (and possibly the edge of the board too), should I think, to be precise, really be defined as follows:
Code:
An eye is a set of liberties of your group which, were your opponent to fill all of them, would cause him to leave his own stones with no liberties (unless the last liberty also happens to be the last liberty of your group, thereby killing it).
From this definition, of course, it easily follows that a group with 2 (or more) eyes can never be captured, no matter how many consecutive plays your opponent gets. Also (of which the above is a corollary) that the final liberty in an eye must always be the last liberty filled (if you're trying to capture a 1-eyed group), hence all the advantages of having an eye in a capturing race.
But, according to this definition, what you had in the corner there was
not an eye! This is because, while one might naturally think of it as a 6-space "eye" (perhaps "pseudo-eye" would be better - note that this is completely different from a false eye), you actually only have 4 liberties there (count them!). This means that your opponent can (given enough plays - as happened here) indeed fill them all without committing suicide along the way, despite the fact you had a genuine eye elsewhere in the group.
So this is why, if the opponent makes an eye inside your "eye", it isn't an eye, because it means that your opponent can fill your liberties without leaving himself with none (as he has the inside of his eye as an unfilled liberty).
This post is, I realise, somewhat in the style of our resident rules authority, which can't be good, so I'll stop here
Incidentally, I think this kind of thing is very easy to miss - at least at my level. I've had games where I've had a group in the corner that I'm sure is alive, and when my opponent played some stones inside still thought was seki, but I later realised (or had it pointed out to me at the end of the game), that my group is in fact dead - because I'd missed that my opponent had (or could easily make) an eye inside
, you need to play 
to stay alive (white will capture, and then it is a seki). Ideally, you would have responded to an earlier white move. If the other player can make his own eye inside your eye, then it's not really your eye. 
Black gains an extra libery and therefore can atari 
