I think the "confusion" is justified. It's a good thing(tm) to start with the concept of "normal with respect to a base" and then move on to say that "normal" or "absolutely normal" means "normal for every base (greater than 1)."

Wikipedia does this, adding the extra term "simply normal" for the case where we are talking about relative to a single base:

    We say that x is simply normal in base b if
    the sequence S(x,b) is simply normal, and
    that x is normal in base b if the sequence
    S(x,b) is normal.  The number x is called a
    normal number (or sometimes an absolutely
    normal number) if it is normal in base b
    for every integer b greater than 1.

For the purposes of explanation, especially to non-technical audiences, I think the approach taken is fully justified.

https://en.wikipedia.org/wiki/Normal_number