To be really picky, GH measures divalent metal ions. In practice, this is mainly calcium with some magnesium and trace amounts of other metals. My son used to work for a water testing company here in the UK and the definition is from him.
And strictly speaking, KH measures the amount of buffer in the water. In practice, this is carbonate. The reason water providers use the term alkalinity is because KH is measured by adding acid of a set concentration until the pH drops to 4.5 (again, info from my son)
Edit to add - it doesn't matter what the cation of the carbonate is, it's still carbonate. And bicarbonate is still a carbonate.