What is the definition of velocity field of a fluid if not the velocity of the molecules of that fluid?

There is only a single velocity field for the fluid at a given time instant, but that paragraph is trying to describe how the acceleration defined as the partial derivative $partial vec{u}/partial t$ is subtly different.

What they are trying to describe is that there are two different reference frames used in discussing fluid mechanics. The first reference frame is called an Eulerian reference frame, and in this frame you describe the fluid as it moves through fixed locations of space. In other words, you act as a stationary observer and formulate the equations for the motion of the fluid moving past each spatial position with time.

In contrast, the second reference frame is called a Lagrangian reference frame, and this is the reference frame used if you imagine a tiny packet of fluid and you formulate the governing equations for how that tiny packet of fluid moves along its trajectory. In other words, you are writing the equations for the motion of a fluid "particle" or "parcel" or "packet" as it moves through space and time.

That paragraph is describing how the acceleration is different in these two different frames, and how they relate. The acceleration in the Eulerian frame is $partial vec{u} /partial t$ and it will tell you how the velocity is changing at every stationary spatial location.

If you were instead interested in the acceleration of a particular fluid packet, then you want the use the Lagrangian reference frame and follow the fluid packet of interest. The acceleration of a single fluid packet along its trajectory is given by the expression provided in your paragraph.

Acceleration in Eulerian reference frame: $$frac{partial vec{u}}{partial t}$$

Acceleration in Lagrangian reference frame:

$$frac{d vec{u}}{dt}, frac{D vec{u}}{Dt}$$

can be related to the Eulerian frame via $$frac{D vec{u}}{Dt} = frac{partial vec{u}}{partial t} + vec{u} cdot nabla vec{u}$$

The derivative in the Lagrangian frame is also often called the "Substantial" or "Material" derivative.

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As an aside, when discussing continuum mechanics, we can't really discuss molecules. The behavior of a continuum is described by equations when we average away all the molecular motion and we're left with some average of how all the molecules within a region behave. Although we often talk about "fluid particles" when discussing continuum motion of fluids, we're really just talking about "an infinitesimal volume of fluid that still contains an infinite amount of molecules such that it can be described as a continuum."

So the velocity field you get from the Navier Stokes equations is the velocity of the bulk motion of the fluid molecules and it is not the motion of any particular molecule in general. It's a minor point that isn't important here, but there is a difference between molecular motion and continuum motion that does become important under some flow regimes.