Measurements - Need some help

[Fonte Boa]

But the snap operation didn't change the measurements... as seen in the first video in the post above at 1:25.

You're sure. But Snap affects the orientation of the longest edges and this affects Intersect operation. Snap, as it was used, deforms the object. The correct would be to rotate that piece so that it reachs the wanted target.

If you make a copy of that object preserving the original
and apply Snap and Intersect in 1st one,
apply just Intersect in 2nd one,
and call Tools|Area & Volume
you will verify they are different.

But what matters here is:
from the moment you Snap those 4 edges (or that single face) to an specific point,
you changed the orientation of the longest edges and,
for consequence, the Intersect results.

The perfect soluction is Rotate to Target, because it doesnt change the nothing any object aspect that can affect Intersect operation.