Relationship between measures of light (candela and lux)

If the source has luminous intensity of one candela in each direction, a type of normalized "candle" ("candela" is a counterpart of one watt per steradian, but with the color/frequency-dependent weights corresponding to the human vision), then it emits luminous flux one lumen in one steradian. One lumen is one candela times one steradian, a unit solid angle which measures the directions from the source. So the luminous flux of a source that has one candela in all directions is $4\pi\sim 12.57$ lumens.

The luminous flux (in lumens) has consequences – it illuminates areas. The larger areas (of the lux meter, for example), the more luminous flux we get. The luminous flux per unit area of the "lux meter" or another absorber is known as the illuminance. The unit of illuminance is one lux which is one lumen per squared meter (i.e. candela times steradian and per squared meter).

If the cylinder were transparent, i.e. if it were not there, the illuminance in luxes would be the luminous intensity times the solid angle over the area. But the area is $\pi D^2/4$ and the solid angle is apprroximately, for $D\ll L$, equal to $\pi D^2/(4L^2)$, so the illuminance in luxes is $1/L^2$ times the luminous intensity (e.g. one candela). Yes, $\pi D^2/4$ canceled.

The same is true if the interior walls of the cylinder are perfectly absorbing; the light going in different directions than to the lux meter won't affect the reading in either case. If the inner walls of the cylinder were perfect mirrors, the illuminance wouldn't decrease with $L$ – it could be calculated as $1/L_{\rm min}^2$ where $L_{\rm min}$ is the short distance between the center of the source and the left side of the cylinder – I was assuming $L_{\rm min}\ll L$.