When researchers first discovered the effects of warfarin, they actually thought that it DID thin the blood. The theory was that 'thicker' blood took a shorter time to clot, so anything that made it take longer for the clot to form must have 'thinned' the blood.
Once they got a better idea of how warfarin ACTUALLY worked, and that it had nothing to do with the viscosity of the blood, the term 'blood thinner' had become a common term describing what warfarin does.
Unfortunately, people use the term 'blood thinner' without really knowing how warfarin actually works. And they maintain this bias no matter hoe many times they're corrected or the actual process is explained.
Unfortunately, we're stuck with this error. Plus, if you say 'anticoagulant,' I'm not sure how many people will understand what that means.
I'll continue to say 'anticoagulant' and correct people (especially medical professionals who should know better) when I hear them say Blood Thinner, but I suspect that many health care professionals (like VitDoc) have an easier time telling patients 'blood thinner' - even if it's wrong - than to try to explain coagulation.