In my opinion, seeing comes before dating, and dating comes before being in a relationship - the relationship being the only one that's exclusive.
"Seeing" someone is like a test run before the real dating phase. When you say that you're "seeing someone" it's like saying that you're feeling them out to see whether you think that you might like them enough to start dating them. You may do date-like activities, but they are much more casual, and often involve other people/mutual friends.
"Dating" someone is not exclusive either, but is slightly more serious than "seeing" (but not by much). In this phase it involves more one-on-one dates, and more "intimate" conversations (i.e. no more small talk, no more talking about the weather). This is the real getting-to-know-you phase, and also the phase where you'll both be evaluating each other like crazy, to figure out whether you match up to each other's qualifications. It either ends in a decision to stop dating each other, or...
"Be in a relationship" - which is when the male tells the female, straight up, "I want us to see each other, and only each other, as a couple". This is the only point where it gets exclusive - nothing before that.
I'd rather label the sections "Seeing someone", "Courting", and "Being in a relationship".. That way less people are confused, because you quoted two of the same thing above, haha.
Regardless, exclusivity is believed to start at "Seeing someone"; but in all actuality it isn't official exclusivity until you have reached the "Courting" stage.
Here's my example of each situation, so you can better reference it for yourself:
1) Seeing someone
- You guys flirt, conversate, hang out, and talk about relationships
- You guys start going on dates, you begin interacting physically (hugging / kissing / etc), etc
3) Being in a relationship
- You guys are official as "boyfriend / girlfriend" and have established a full relationship level of comittment.
When they update status on Facebook or MySpace, when you have the talk, ...