What is friendship?

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This question is quite fundamental, and I'm going to split it into a short question and an explanation.

Short question: How do you know if somebody is your friend?

Explanation: I've been posing this question to myself since kindergarten, where I first heard about this word. Back then, the other children were rather explicit about friendship, saying things like: “Let's become friends!” They didn't say that to me, however, and I had no idea what they were talking about. Concerning the years at school (age 7–19), it's pretty safe to say that I had no friends at all. There were some people with whom I shared some interests and participated in some activities, mostly music; but I don't think this meets the definition.

Things turned better when I entered university (age 21). First, the increase of group work was not a bad thing. Unlike at school, students usually didn't pick their coworkers by sympathy but by topical skills – and it's quite safe to say that I'm fairly good at math. So I almost never had any trouble finding someone to work with. But OK, doing university homework with someone doesn't qualify as friendship, either.
The key was, again, music. I quickly joined the university orchestra. As ever before, I enjoyed the music, which would already have made it worthwhile. But after a couple of months, two other math students who also played the violin invited me over to a barbecue party. Although I was a bit scared because I hadn't been to so many parties before, I showed up, and this has been the best decision of my life so far.
The current status is: I'm “friends” with these two math students and one CS student (who plays the cello), which means they have been inviting me over regularly for the last 1.5 years. This probably meets the technical criteria of friendship. On the other hand, Wikipedia, quoting the Oxford dictionary, says: Friendship is a relationship between two people who hold mutual affection for each other. This is what I'm not sure about. Although I meet them so often, I still have the feeling that there is something missing. That is, I feel some kind of “barrier” that stops me from forming a close emotional relationship.
The other thing is: I primarily take part in the activities they propose. I'm infinitely grateful to them for letting me do so, but it's still not what I would expect from a friendship. Whenever I propose something, it gets rejected rather quickly, but on the other hand, they say things like: “You're so quiet and so shy.” So it's probably not their fault.
I should probably invite them to my place sometimes, but my apartment is really small, and I'm afraid of making social mistakes.

So I'm really not sure: Do I have friends?

Knowing the answer to this question is not only interesting, but I think I also need it to get rid of the “barrier” mentioned above. Part of this barrier is probably my (unconscious) “NT-mode emulation mask”, but the other part is that I can't approximate someone further if I don't know about the current affectional status.
This leads to my original question: How do I know if I'm friends with someone?
If I know someone well (which may take more than one year), I know if I like him/her, but I have no idea how to tell what he/she thinks about me. My “potential friends” that I talked about above sometimes casually tell me that I'm their friend. I could accept that and say: “If they tell me we're friends, they are probably right.” But I'd prefer to work that out by myself. That is, I want to be able to tell what kind of affection they feel towards me.

Every person in this world defines friendship in a different way. For some, what you describe is friendship. For others it isn't.

I happen to have the most strict definition of friendship - someone who is always there for you and you for them, with whom there's full mutual trust and who appreciates you for who you really are, without games. Someone who is as interested in the relationship as you are, where one is not more powerful in the friendship than the other.