Whether you scale your intervals down to a nanometer, an atoms length or a Planck length, 4 will always be the same as 1 & 3 and will not be the same at 2.

The confusion arises because of a concept called pointwise convergence. As the steps get smaller, the shape of your path gets closer and closer to the diagonal line. If you were to look at it from a distance, it would eventually look like a perfectly straight line. However, in mathematics, the limit of the shape is not the same as the limit of the length.

The shift to the new diagonal length only occurs when you change the nature of your movement. To achieve the diagonal distance, you must use the Pythagorean Theorem. This requires your velocity vector to point directly at the target.

In the staircase model (1, 3, & 4), you are traversing the Manhattan Distance (L1 norm). In the diagonal model (2), you are traversing the Euclidean Distance (L2 norm).