What is the correct definition of a group?

Definition A is the correct interpretation.

A group is a pair $(G,+) $ where $G $ is a set and $+$ is a function from $G\times G $ to $G $ satisfying certain properties.

Perhaps confusingly, the group is also called $G $ (often). So two different entities -- the group, and the underlying set -- may be referred to by the same name. For example, if someone says "$g \in G $", then here $G $ is referring to the underlying set. It would be too laborious to use different names for the group and for the underlying set.

You appear to be asking whether on one hand a set is a group, if an operation with the correct properties exists, or on the other hand whether the group comprises both the set and the operation.

The correct definition is that the group is set together with the operation.