Yes it is...if you work freelance, you're a freelancer, when you work, it's called freelancing.
There's a massive difference between doing a bit of extra work every so often and being a freelancer. I always find developers in full time jobs have ridiculous ideas of what freelancers earn, it's simply not true in most cases.
I used to freelance, and moved onto fulltime, and it did not involve a single agency, infact I didn't even need to leave my house. www.sitepoint.com, www.freelancers.net, and others for work. Had me earning £100+ ph on some jobs. I had a set price of £500 for a typical e-shop setup, which I would have done in less than 5 hours with ease. You are referring to contracting, not freelancing.
Hence my mentioning of templates (pre-made) and existing CMS etc. If Jim's Spanners Shop just wants an eShop with a template, why would I create an entirely bespoke site? £500 for a site to Jim is not a lot, and will very quickly repay its self.
And yes, both of your definitions fit the dictionary definition...but again, you're the one that said someone working on-site wasn't a freelancer. Where did I claim they were entirely different?