My instructor didn't actually call it a slip pivot, but the description puts the slip pivot (one step, according to the definition I got from the USISTD book) on step 6, not step 4: steps 4-6 for the man are side, back, back, turning 1/4 L on step 6 with L foot held in CBMP. However, our instructor did have us go into fallaway on step 3.
The slip pivot as a component could be thought of as consisting of one or two steps, depending on whether you include the initial step into fallaway position. There is no single, correct answer in this regard. I can see why ISTD might define it as 1 step: The Fallaway Reverse and Slip Pivot figure consists of a fallaway reverse, which most would think of as 3 steps. That leaves you with 1 step, the slip pivot. But if you think of a Chair entry, the answer is more ambiguous: Is a Chair a single checking action, or does it include the return to the original foot? That, too can be thought of either way.
It is expected that whole figures will often (but not always) contain more steps than the raw components that comprise them. Some figures consist only of one single step or component (e.g. Reverse Pivot). Some consist of combinations of two or more (e.g. Spin Turn = 1-3 Natural + Spin Turn component), while others consist of a component with a step or two before and/or after, in order to provide context. The name of the figure and the component may be identical, so where there is question, one must clarify, e.g. "dance just the 3-step Twinkle itself, not the whole 6-step figure". In fact, the same is true of components themselves, as is the case with a chasse, or a chair, or a slip pivot.
Our slip pivot as a figure in American style consists of 3 steps. The first step (or two steps, depending on your point of view) are the context, and the last part is the raw component. As a figure, it's still much more granular than the old Fred Astaire Fallaway and Rock, being only 1 measure rather than 4.
There are pros and cons to mincing up your syllabus into smaller bits. The best argument in favor of more granularity is that it affords the student more options for composition. To follow the Fred Astaire version, you would need to dance the same 4 measure sequence in exactly the same way every time. With the figure narrowed down from whole combination to simple component with minimal necessary context included (i.e. 3 steps total), it opens you up to a world of possibilities.
Now that's not to say that the more we break things down the better. A syllabus would start to look pretty silly and pointless if figures were broken down into all single-step actions. But it's my belief that the majority of American style syllabi, especially in higher levels, err on the side of ridiculously long combinations of choreography. So we've made the smaller figure approach a staple of our syllabus.
To that end, it makes sense for us to start the figure in closed rather than fallaway; If we'd felt it necessary to begin in fallaway, it would have been more correct to make it a 2-measure figure and include the preceding box that ends in fallaway position. That would in turn rob it of much of its flexibility with regards to precedes. So as far as syllabus construction, it was a better choice to use the version beginning in closed and turning immediately to fallaway on step 1, even though in a vacuum either choice could have been correct.
However, when I said previously that I felt a closed entry is "stronger", I wasn't referring to syllabus construction. I actually think it produces better results from the standpoint of the dancing. Fallaways of all kinds can be danced with the lady's head either open or closed, but those culminating in slip pivots (e.g. Fallaway Reverse & Slip Pivot) are typically danced with the lady's head closed, and for good reason: The pull of the lady's head weight to the left assists with the action of the pivot.
