Is the Reynolds Number well defined for a given system? I have been looking for definitions of the Reynold number, and by far the most common definition is that the it is given by:
$$ Re\equiv \frac{LU}{\nu}$$
Where $L$ is the characteristic length scale, $U$ is the characteristic velocity scale and $\nu$ the kinematic viscosity. 
To me this seems to be a very imprecise definition, since the characteristic length scale and velocity scale don't have any precise value for a system and are free (within reason) for the experimenter to chose. This  surly means that $Re$ does not have a set value for a given system but can take any value (again within reason).
Am I correct in saying this? If so is there any other ways to define the Reynold number so that for a given system it has a set value that all experimenters would agree on and if not why not?
 A: Reynolds Number is not well defined for a system rather it is well defined for a given state of the system, i.e: Reynolds number describes the steadiness of the flow of a fluid in the system.
A simplified form of the equation you mentioned as given in High-School Physics textbooks goes as  (essentially the same as what you said)
$$Re =\frac{\rho ud}{\eta}$$ 
where $\rho$ is the density of the fluid, $u$ is the flow velocity and $d$ could be any geometric parameter which describes the area of flow (diameter for a circular pipe)
If you look at the numerator closely, it consists of density, flow velocity and the diameter of the pipe. An increase in any of these parameters would cause an increase the total inertia of the fluid. More the density, the stronger the fluid will oppose to changes! Larger the velocity, lesser is the tendency to adjust to its surroundings. The larger the area of cross section, lesser is the  
If you now look at the denominator, it is controlled by just one term, that is the coefficient of viscosity. The larger the value, the more are the viscous forces. The more the viscous force, more is the tendency of the fluid to adjust to its surroundings.
Combining the above facts, you can consider the Reynolds number as the ratio of the inertial forces to the frictional forces. 
The more inertia the fluid has, the more is its tendency to distort the flow and hence the more turbulent is the flow.
The more is the power of the frictional forces (viscosity), lesser are the distortions in the flow. 
Hence a larger Reynolds Number implies that the flow is turbulent.
How large? There is no definitive value or a margin or cutoff after which a flow is said to be turbulent. Generally, we say that the fluid has some turbulence if the Reynolds number is larger than 500. We say the fluid has turbulent flow if the number evaluates to be more than 2000. A very small number like 10 would mean a steady flow.
Reynolds number is a mere number which gives an intuitive idea of how turbulent the flow of fluid is. There is no requirement that every experiment yields the same value of Reynolds number since its very sensitive to the change in parameters. 
Your view is absolutely right. 
There simply can't be a number which serves the purpose of introducing Reynolds number where all experimenters can agree on because the steadyness of flow depends on both the velocity as well as the area of cross section. 
A: You are right in saying that for the same flow problem there is no set value of $Re$ that everyone is compelled to arrive at. Usually people look to initial/boundary conditions of the problem to choose scales for length and velocity. Say you are studying steady, fully developed, laminar flow in a long pipe. Pipe's radius appears in no slip boundary condition, and since the flow is fully developed it is the only length scale that appears there. So that may be the favored choice for length scale in defining $Re$. But if someone decides to go for pipe diameter that is fine too, as long as he consistently maintains that definition throughout his analysis. Similarly one may take average flow speed for the velocity scale in defining $Re$, but if someone else decides to take maximum velocity instead, that is fine too, provided again that he maintains that definition consistently throughout his analysis. Therefore there is no way to uniquely define $Re$, because one may always work in terms of some non-zero scalar multiple of any one definition of $Re$, and there is nothing wrong about it provided (again) that he sticks to that definition throughout his analysis.
This lack of uniqueness is a nuisance in practice because you must always check other people's definition first. However all definitions are inter-convertible (in principle at least), so there is nothing fundamentally wrong with adapting any one definition you like. Problem occurs when you chose a scale for length or velocity that is not relevant to problem at hand. In the fully developed flow example, distance from entrance of pipe is not a relevant parameter because the flow has forgotten its past and achieved a steady state. If someone were to adapt this as his length scale to define $Re$, then you may say that definition is wrong (for this particular problem).
