There are two separate issues to consider here.
First, a path along the direction of the vector 2 has to traverse a greater distance than that for vector 3 to affect the same change in the value k. Hence, the vector 2 direction could not possibly have greater rate of change. Then, you could consider if there is another vector in the 3d space that defines a direction of greater rate of change than that of vector 3. It doesn't seem that there is.
The second issue is much more important. That is, it is a mistake to even be thinking of 3 dimensions when the 2D gradient is being considered. A gradient is an operation performed on a scalar function, and the dimensionality of the domain of that function determines whether a 1D, 2D or 3D gradient is being considered. A 2d gradient in xy-system does not know anything about the direction z.
However, there is another important point here. In a sense, there is a mapping from the 2D xy-plane to the 3D surface, but the surface, which abstractly exists in the 3D space, is still a 2D structure. Note that, because of this mapping, as you travel along the direction of the vector 3, you are also traversing along the direction of the vector 2, due to this mapping. So, in some sense your visualization is not wrong. However, rate of change, or gradient, or any of the other vector space notions are things to consider only in the xy-plane.
It's important to understand this, so let's try to put a physical example in place. In a 2D electrostatics problem, you might determine the potential voltage V as a function of the coordinates x and y. Hence V(x,y) is something we can take the gradient of to determine the electric field. That electric field will have direction as a function of position (x,y) and will define directions only in the xy-plane.
One can extend the example to a 3D problem. Then, we have V(x,y,z), which will define electric fields that have directions in 3D space. In this case you would not even be able to visualize a fourth dimension to create a vector analogous to your vector 2.