If you starting out in the murky world of internet dating, you should be aware that the rules of traditional dating don't apply. Keep safe and only meet in public places. I think it is ok to ask someone out if you feel there is a connection.
Exclusive is when you both feel that you have enough of a relationship and prospect together that you don't want to look around for anyone else. You cannot assume someone you meet on the net, or elsewhere, who declares exclusive intent isn't dating around...it is a funny old world. Take things slowly and get to know someone before parting with the heart strings. You also need to think about your relationship with your ex, and how you explain that to a new partner. I am close to my ex for the sake of the kids, and although we are not FWB I have discovered that new suitors have always viewed him with resentment and mistrust.
I've never met someone on a dating site but I would assume there's several schools of thought: 1. they aren't seeing other people (i.e. we're exclusive immediately) because the whole point is you're trying to find a relationship 2. no-one wants to risk not meeting "their soul mate" so want to date a bunch of people before fully committing (i.e. not exclusive until the talk happens) 3. they're only on the site "for fun" and not anything serious 4. that unless it's explicitly said one way or the other, that you think the same as them (which is any of the 3 above)
Basically, you need to have that chat on the first date: "before we go any further, I just want to make sure we're on the same page... at what point do you consider this exclusive?" and they'll ask your opinion too.
FWIW I'm of the "unless we've had the exclusivity talk, we're not exclusive" school of thought... which did get me into trouble once because this guy I kissed (yes, just kissed) seemed to think that meant we were now dating and exclusive and it really upset him when he realised we weren't.