Yeah, strict vegetarian is essentially the same thing as vegan
This is just my opinion and experience. I'm not vegetarian, but I know several people who are. People who are/practice Vegan tend to know it. Vegan eschews all animal-based products. No milk, no cheese... nothing which could possibly be traced back to animal origin. More general "vegetarian" tends to come in shades, depending on why the person is vegetarian or their understanding of food origins.
Basic vegetarian may not eat meat because of the harm done to the animal, but will usually eat byproducts such as milk-based products. Eggs may or may not be allowed (since they're never fertilized, they aren't really an animal), and sometimes fish are allowed. I've never really understood why some vegetarians let themselves eat fish...
I've also rarely encountered people who are struggling with beginning a vegetarian lifestyle, and will consider themselves vegetarian while occasionally slipping in a bit of meat. Often accidentally in compound forms like fries at McDonalds (they use beef-based oil, you know).
"Strict" vegetarian is a fairly common term, but it's meaning isn't consistent. Usually I find it means people who won't eat eggs or fish, but still eat dairy products. But opinions vary...