Oh calm down. I've done most of those things, too- and my training was strictly on the job. I was referred to as a technician, but I'm fine with being called an assistant too. The things I didn't do (from your post) are suture and pull teeth. I'm not sure those things were legal where I worked, but who knows ...maybe the docs just preferred to do those things themselves.
Vet techs do work under the direct supervision of a veterinarian, though ...like medical assistants. The tasks themselves can be very different, but that's not what I was commenting about.