Wouldn't the gravitational wave distort space itself and everything in it?

Perhaps I'm not thinking correctly, and I'm sure the folks working on this are far smarter than I.
I would expect you can only observe this from the perspective of a higher dimension.
The fact that space is warped by a gravity wave is not "known" to the light beam.

Imagine a perfect 3-D Euclidian space, now put an observer in that space with a laser or a full blown LIGO.
Observer fires the laser and the light travels perfectly straight and covers a specific distance in a specific amount of time.

Both you and the observer agree on this.

Now bend, twist, or scrunch up that perfect 3-D Euclidian space, making it more like hyperbolic geometry. This is the effect we expect from a gravity wave, yes?
When the observer fires his laser this time, he sees the light travel perfectly straight and covers the same "distance" as before.

But you would disagree. From your "outside" perspective, you would say the light did not travel "straight" this time, and that the light traveled a shorter distance, if "space" was scrunched up, or a longer distance if "space" was stretched.

How then, can we possibly detect a distortion of space when it is the same space we occupy? We are the "observer" above and LIGO will always tell us the light in each arm is traveling the same distance, even if one arm is distorted by a gravity wave.

What am I missing (besides a degree in physics)?