Let's start with a few baseline statements about boundary-layer stability and transition:
(1) In general, there is no critical Reynolds number where laminar flow becomes turbulent. You can apply some simplistic ideas like that to pipe flows and a handful of other special cases, but even then, the underlying physics involve the growth of small disturbances until they become large and break down to turbulence. In other words, even in a pipe flow, passing that critical ##Re## doesn't immediately mean turbulence. It means the flow is unstable and will soon be turbulent.
(2) The comment in Landau & Lifschitz appears to be describing the concept of the minimum critical Reynolds number, ##Re_{cr}##, which is the smallest ##Re## at which small disturbances to the base flow becomes unstable. It should be noted, however, that this does not imply that turbulence is inevitable; it only means that there is at least some degree of instability present, but three may not be enough growth to lead to transition in a given situation. ##Re_{cr}## used to be the state of the art in boundary-layer stability and transition. These days, it is long-deprecated (though still an important piece of the puzzle sometimes).
(3) Consider the generic definition of ##Re##
Re_{\ell} = \dfrac{U \ell}{\nu}.
Also, for illustrative purposes, consider the Blasius boundary layer thickness,
\delta \propto \sqrt{\dfrac{\nu x}{U}}.
So, if we define a ##Re## based on this reference length, we have
Re_{\delta_r} = R = \dfrac{U}{\nu}\sqrt{\dfrac{\nu x}{U}} = \sqrt{\dfrac{Ux}{\nu}} = \sqrt{Re_x}.
What this says, then, is that ##R## is equivalent to the square root of the ##x## Reynolds number. If you look at what Landau & Lifschitz did, their ##A## parameter is related to this concept (though not identical). ##R## is a very common parameter in the study of stability.
One general comment:
(4) Subsonic flows are governed by elliptical equations, and as such, what occurs at one location in the flow field affects the rest of the flow field. The influence of a body does die away at a distance, but it is never truly zero.
And some comments on vortex shedding in light of the above:
(5) While vortex shedding does not have a whole lot of shape-independent rules of thumb, it is generally a result of boundary-layer separation. This also means that it is inextricably linked to the idea of laminar-turbulent transition.
(6) For example, consider a cylinder. As ##Re## increases, the boundary layer separates and leads to large separation vortices. As ##Re## continues to increase, these trailing vortices begin to shed, creating the familiar vortex street with laminar vortices. At higher ##Re##, the boundary layers transition, causing the separation point to move rearward, the wake to shrink, and the vortices to stop shedding. At a higher ##Re##, the now-turbulent vortices begin shedding again. So, as you can see, there is no basic principle for a vortex shedding ##Re##.
(7) The above example was for circular cylinders (and spheres). There are infinitely many possible shapes that will experience vortex shedding, and they will all have different characteristics. The reason that ##R=\sqrt{Re_x}## is relevant is that it is a good parameter for correlations with laminar-turbulent transition, but it is not a hard and fast rule. Even on a subsonic flat plate with zero pressure gradient, we can't tell you what ##Re_{tr}## is in general, let alone more complicated shapes.
