"cane sugar" is cane sugar, because sugar can come from a few sources, including cane and beet. Most of the white sugar you know is either from cane or beet and it's generally glucose. Another sugar, fructose, is fruit sugar, or grape sugar, which is dextrose - they are all "sugar".

Indicating the (near) exact type of sugar is not a bad idea.

Sugar beets account for about 20% of sugar production, so it's not unlikely you've had non-cane sugar. Still no real reason to label it as cane sugar though.

It's not. A lot of sugar comes from sugar beets

The vast majority of sugar in the US at least is from corn.