No he didn't, he left out length contraction. To get the difference in the amount of proper time it would take to make the journey you need to work out out the difference in coordinate time over the distance in space.
Length contracts by the same amount as time dilates so it should take 270 years proper time.
You were right to highlight who measures the proper time and who measures the proper length as it one of the most important things about special relativity.
Non proper length is measured by the spacecraft, because the spacecraft is measuring events that occur in different places with respect to the spacecraft (the spacecraft could be said to be observing moving planets). In the example this gives the spaceship distance meter calculated reading of 264.23 light years from planet to planet. Non proper length is length contraction, shortening of the distance and is designated L.
Proper time is measured on the spacecraft because its clock is at rest with respect to the spacecraft; a time interval measured by a clock which is at rest relative to the observer. In the example the spacecraft clock gives a journey proper time calculated to be 377.47 years from planet to planet and is designated ∆t0.
The spacecraft instrument dashboard would show total journey distance to be 264.23 light years, total journey time to be 377.47 years at a speed of 0.7c. Put these figures directly into the equation: Time = Distance/Speed (t=d/s) t=377.47years, d=264.23 light years, s=0.7c
377.47 years = 264.23 light years/0.7c and it all makes sense.
Proper length is the length (distance) measured by an observer (on the planet in this example) between two points (planets), at rest with respect to them. In the example the proper length planet to planet distance measurement is 370 light years and is designated L0.
Non proper time is measured by an observer on the planet in this example, because the events are occurring in a different place (the spacecraft in this example) and the clock on the planet is not at rest relative to the moving spacecraft. In the example the planet clock gives a journey non proper time calculated to be 528.57 years from planet to planet and is designated ∆t.
The observer on Earth would have the following control panel readings:
Total journey time=528.57years, total journey distance=370 light years, craft speed=0.7c
Put these figures into t=d/s gives 528.57 years=370 lightyears / 0.7c and it all makes sense.