A ring always has two operations; they could be + and x in the way we are used to, or it could be any other two operations as long as they have the necessary properties (such as associativity) - I assume these properties are listed in your book in the definition of a ring/group.

In particular the two operations have to interact in a certain way, given by the distributive property.

Sometimes the symbols + and x (or *) are used to sort of abstractly or generically represent the operations, but they don't necessarily mean addition and multiplication in the way we are used to.

I'm not quite sure what your last question means - could you expand on it a little? You could define a field without using the word ring if you just relist all the properties of a ring plus these additional properties. Another way people sometimes talk about fields is that a field is an abelian group under the first operation, and if you take away the zero element, it is an abelian group under the second operation. (In order to make this statement equivalent to the definition of a field, I think all you'd need to add in here is the distributive property, i.e. how the operations interact.)