This is a terminology question. Answers will help me satisfy a referee but I'm also genuinely interested. Consider the following two things that you could define for a topological space X:

(1) The Grothendieck ring of the category of real vector bundles over X;

(2) the set $[X, \mathbb{Z} \times BO]$ of homotopy classes of maps from $X$ to $\mathbb{Z} \times BO$ where $BO$ is the classifying space for the infinite orthogonal group.

It is well known that (1) and (2) are in bijection when $X$ is, say, a finite CW-complex. This can fail when $X$ is not compact though, and I think it also fails when $X$ is a nested sequence of circles in the plane with the subspace topology. Anyway, the two are different in general.

Now consider the following names and notation :

(A) the real K-theory of $X$;

(B) the real topological K-theory of $X$;

(C) the K-theory of real vector bundles over $X$;

(D) $K(X)$;

(E) $KO(X)$;

(F) $K_\mathbb{R}(X)$.

Which would you pair to which? and what is your favorite for (1) and for (2)?

For example I don't expect anyone to answer that C2 is reasonable. I'm curious whether E1 or E2 will appear. Some other notation may also be prefered, I think I saw $\mathbf{KO}(X)$ once, meaning (1), while $KO(X)$ was for (2). Or perhaps the other way around. There are also alternative names like "the Atiyah real K-theory", not sure meaning what.

Thanks! There may not be any universal convention, but at least we can have a vote of sorts.