For instance, take the example of this book here on page 30. About the formula $$ Q(x,x') = R(x), S(x,y), S(x',y), R(x') $$ described there, what do the commas mean? Is it the same as an "and" operation? Or a natural join maybe?
The notation is defined on page 17 and 18. The authors define a couple notations and state that they will "freely switch back and forth" between them. This example doesn't seem super-consistent, I'd expect $Q(x,x'):\!\!-\,R(x),S(x,y),S(x',y),R(x')$ rather than an equals sign, but presumably it means $Q(x,x') = \exists y.R(x)\land S(x,y) \land S(x',y)\land R(x')$.
It doesn't really make sense to talk about natural joins in this context, as those rely on named attributes/columns. But yes, this query includes (equi-)joins. To be clear, the comma stands for conjunction. Any joining is implicit in duplicate uses of a variable. In particular, if all variables were used only once in the right hand side, then there would be no equi-joins, even implicitly. In SQL, the query would look something like:
CREATE VIEW Q AS SELECT DISTINCT S1.Field1, S2.Field1 AS Field2 FROM S AS S1 INNER JOIN S AS S2 ON S1.Field2 = S2.Field2 WHERE S1.Field1 IN (SELECT Field1 FROM R) AND S2.Field1 IN (SELECT Field1 FROM R)
or to be more systematic,
CREATE VIEW Q AS SELECT DISTINCT S1.Field1, S2.Field1 AS Field2 FROM S AS S1 INNER JOIN S AS S2 ON S1.Field2 = S2.Field2 INNER JOIN R AS R1 ON R1.Field1 = S1.Field1 INNER JOIN R AS R2 ON R2.Field1 = S2.Field1
or going a different direction,
CREATE VIEW Q AS SELECT DISTINCT S1.Field1, S2.Field1 AS Field2 FROM S AS S1, S AS S2, R AS R1, R AS R2 WHERE R1.Field1 = S1.Field1 AND S1.Field2 = S2.Field2 AND S2.Field1 = R2.Field1