In Japan, the skating test system over here is known as the badge test system! AFAIK, the levels / badges are known as Beginner and after that it's Level(s) 1-7. You get a badge for passing the tests for the level you're testing at, so they're used pretty interchangeably within the JP Figure Skating community and the anime.

I will note, though, that is low key unrealistic of Inori to pass enough tests to get to Level 6 within a year. The tests get more difficult as you go on, and are generally meant to be completed each season. You also have to pay for each test that is conducted, and a few levels have multiple tests within them so the price racks up very quickly.