Larson initially composed the postulates from seven assumptions, I believe, and later decided, for the sake of conciseness, not to explicitly include those that were clearly implicit. He observed that at least one of these implicit assumptions, could later be expressed explicitly, if omitting it caused confusion.

However, the fact that you infer from the postulates that "the term 'space' is actually referring to a distance measure," indicates the difficulty inherent in clearly, yet succinctly, expressing the assumptions. The term "distance" indicates a span of space measured by a velocity over time. Thus, ds = v*dt. However, we can also indicate the same measure of space in terms of an interval of time. Thus, dt = ds/v. This fact reveals that velocity may be interpreted as a meter of space and time wherein it, in a sense, creates space (distance), given time, or time (interval), given space.

However, there is another interpretation that, although subtle, reveals an important distinction. This interpretation is that the equation v = ds/dt states that motion is equivalent to a ratio of a change in space to a change of time; that is, they are the same thing. Given this interpretation, motion cannot exist except in terms of this ratio of changing space and time. The first fundamental postulate posits the existence of motion, as the sole constituent of the universe. Therefore, it posits the existence of its equivalent, a ratio of changing space and time, as the sole constituent of the physical universe.

Thus, your second postulate, that posits "motion expressed in space" presents a conundrum, similar to the one we see arising from the conflict of GR and QFT: how do we describe motion in space, when space is defined as an aspect of motion? It's like trying to write a wave equation for gravity that must evolve over time, when time is an aspect of gravity! There's a built in contradiction stemming from the definition of things.