farley: No, not a published definition, just my understanding of a forward slip; and no, what you're describing is not a slip at all. in Canada, we call it a crab, when the relative wind is still in line with the plane, your track is in line with the runway, but your longitudinal axis is shifted into wind.
I think the Flight training manual basically emphasizes the difference is in the PURPOSE of the slip.
Lose altitude: forward slip
Change your track (or maintain a straight track in x-wind) without changing heading: side slip