Just wanted to confirm this is perfectly normal for two to three year olds, they all do this, including the articulate ones, and one just gets through somehow and out the other end.
I don't think it is a problem your DD saw you cry, not that she is old enough to really appreciate it, but it shows her that what she does has consequences for you and she can upset you. More importantly is not giving in - because once you do that, then she learns that tantrums are an effective manipulation technique, and will keep on with them even past the age when they normally hit (which is perfectly rational behaviour, if you engage in behaviour that gets you what you want, why would you not continue?). If you don't give in, then eventually (and I mean eventually) they learn that this is not effective (especially if they work out that other techniques, like asking nicely, being good or negotiation are more likely to work), and they grow out of the tantrums, but that takes time.
What works for tantrums, in my experience, varies from DC to DC. I did sometimes find just hugging tight during the tantrum was effective, depending on whether that was appropriate. For one DS, for example, he was distractable, and after giving him a few minutes for the worst of it to blow, I would try distracting him with something else. With DD, however, who was incredibly focused and could never be distracted onto anything, the technique that worked best was to make her laugh, and the most effective way of doing that was often for me to "play imitate" the tantrum - so if she stamped her feet, I would stamp my feet in a very silly clowning way. If she lay down and screamed, I would lie down (obviously not possible if out), and pretend to kick my hands and feet in a play silly imitation of what she looked like, and she often found it so funny to see mummy do this that she would laugh and snap out of the temper tantrum. Can't promise it works with anybody elses DC, just something I worked out with this particular one. Point is, she didn't get whatever it was that she wanted, but if what she really wanted was mummy attention and getting down to her level, she got that. Obviously I was not in any way, shape or form, actually temper tantruming, I was just making the gestures that, if you think about it, are really rather amusing, in an exaggerated form.
And they do grow out of it, she won't be doing this when she is applying for university!