That sounds like a class.
Having the functions that operate on the struct attached directly to the struct declaration, vs having some functions that the first parameter is the struct on which the function operates, doesn't seem like a particularly meaningful distinction to me. OK, you like C-style programming in favor of C++-style programming, congrats. It's still a class either way.
Enums are are the better example of non-class types. For example, you can have:
enum StringOrInt {
String(String),
Int(u32),
}
And you can go ahead and implement methods on that type. Classes have "AND-state", not "OR-state". But a Type in general can have either kind of state.