Condition for aliasing
There really is no practical difference between the two. The important thing to understand about aliasing is not the exact definition of the word, but rather the concept: when two frequency bands alias and there is a signal in one of these bands, after sampling you can't tell which band the signal came from. In a sense, by the time you're asking what is or isn't aliasing, it's already too late.
Consider this: you have a digital audio recording of perfect silence. However, somewhere during the recording process some RF-interference was mixed in. So the recording of silence now contains some aliased RF-noise, which isn't added to any existing spectrum, because there is none. After all the recording is silence, even though clearly the speaker cone is moving. Maybe the lack of spectrum was actually a feature of the spectrum?
If you understand the concept of aliasing, then it is almost always clear from the context what the exact definition in that particular context happens to be.
Edit:
I want to clarify further that what the text is really trying to say is simply if you violate the sampling condition, then the bands will alias and if the bands alias, then the signals can alias as well (but when they do, you really can't tell; at that point it's too late).