You seem to be handling units fine. Both formulas are based on the equation V*t=ΔB*A*N. But these two approaches suggest different ways of estimating V and t. For example, the LT version refers to the secondary side, and uses the assumption that the converter is operating at steady state and continuous conduction mode, so Vs*t=Vs*D/f=Vout/f. The second formula takes a much more conservative approach, which assumes that the maximum duty cycle and input voltage may happen simultaneously (plausible under transient conditions, implausible at steady state). So the second equation will give a much higher number of primary turns than the first.