In terms of diet, am I a vegan or just a vegetarian who can't eat eggs, dairy or honey?

There is no such thing as a vegan diet, a vegan consumes a strict vegetarian diet but as Free and others said veganism is a lifestyle. But to be a strict vegetarian you need to be checking the labels of any packaged foods you buy for not only dairy and eggs, but also hidden ingredients like the red colouring number 120 or E120 (called cochineal or carmine) which is made by boiling the wings of the cochineal mealy bug. This is just one example, there are dozens more of these types of ingredients which appear only as numbers in the ingredients list. So unless you avoid these as well, you are a plain old vegetarian but that's fine- you are still doing more than about 95% of people in the world to reduce suffering and environmental degradation.