Are vegans vegetarians? and related questions

I recently asked the following questions:

Assuming all other requirements are the same:

1. Is a vegan a vegetarian?
2. Is a vegetarian a vegan?

To my (somewhat) surprise, this drew quite a lot of interest from colleagues, friends, and even people on Facebook I hadn't talked to for years, with lots of different ways on how to interpret the questions.

Restaurants

The next two questions were:

3. Is a vegan restaurant a vegetarian restaurant?
4. Is a vegetarian restaurant a vegan restaurant?

Apples and labels

I asked two more questions:

5. Is a bag of apples a bag of fruit?
6. Is a bag of apples that is labelled "Apples" a bag of fruit?