What is a precise definition for laminar flow that includes "laminar vortex-shedding"? I can't find a definition for laminar flow that includes vortex structures. 
See paper: Numerical calculation of laminar vortex-shedding flow past cylinders
and countless others when googled.
 A: 
I can't find a definition for laminar flow that includes vortex structures.

Vortex structures don't figure directly in the definition of laminar flow, in the sense that we can have laminar flow with or without vortex/eddy structures. Laminar pipe flow doesn't have vortex structures (it has "vorticity" which is different), while vortex shedding at low enough Reynolds number is an example of a laminar flow with vortex structures.
The distinction between laminar and turbulent flows is that in a turbulent flow vortex structures occur in a variety of sizes, from the smallest Kolmogorov scale to the integral scale, and also in a variety of shapes (blobs, sheets, tubes, ribbons). But a laminar flow doesn't have such a multiplicity of scales and structures; vortex blobs being shed behind a cylinder or sphere in laminar flow are approximately all the same size (if the Reynolds number is small enough these blobs don't become turbulent after being shed but simply dissipate away). This is a qualitative idea and making it precise requires use of Fourier transform and velocity correlation functions; see Turbulence by P.A. Davidson.
Also there are no separate equations for laminar and turbulent flows; the same Navier-Stokes equation presumably gives rise to both kinds of flow. However when dealing with turbulent flows we often use the averaged form of Navier-Stokes equation (called Reynolds-averaged Navier-Stokes equation) but the resulting equations are not closed (more unknowns than there are equations); so we must adopt ad-hoc models to obtain closure and there are plenty such models. However Navier-Stokes equation can be solved without averaging, and therefore without requiring ad-hoc models, and these simulations are called Direct Numerical Simluations (they are computationally expensive).
