The way I learned how to carve was kind of more intuitive. I had teachers tell me to do this with my body, this with my legs, this with my butt, this with my hair, but I was never really told what a true carve was. What my board was supposed to feel like when in a carve?
In a proper carve, you're shifting from edge to edge and don't lose much speed. You're essentially riding your edge instead of scraping the snow. You'll still displace snow so when you're going quick, you still get a cool spray.
I basically stopped scraping my edges through the snow, and more of riding the edge and pushing just hard enough not to "scrape" the snow. I liken it to driving where your carve is when you have tire grip, and when you scrape is when you lose traction. My legs pushing is like the throttle. Push hard enough so you get that nice S line and remain in control (grip), but not too hard that you scrape the snow and dump speed (no traction!).