What do the L, M and S pixel values in the LMS color space stand for exactly?
Colors specified in CIE LMS don't have more or less meaning than colors specified in other CIE based color spaces. CIE LMS coordinates can be readily transformed between CIE XYZ, CIE LAB etc using standard matrices.

However some applications are more suited to some color spaces than others, and LMS has its uses. 

In LMS color space, the color matching functions are very close to being the sensitivity functions of the human LMS cones (cone fundamentals). Since each dimension of LMS color space relates to a single cone, it makes it easy to model some of the per-cone gain adjustments that happen in the retina.

cone sensitivity functions:



Basically the retina adjusts the sensitivity of different cones when trying to white balance a scene and some digital imaging white balancing algorithms mimic these operations in LMS space.

This is the paper on CIE LMS color, also known as the '2006 Physiologically Relevant Color Matching Functions' by Andrew Stockman
http://cvrl.ucl.ac.uk/people/Stockman/pubs/2006%20Physiological%20CMFs%20SS.pdf Though it mostly talks about the derivation not the use cases of LMS.

This chapter is also by Stockman and has a lot of discussion on the cone fundamentals http://color.psych.upenn.edu/brainard/papers/Brainard_Stockman_Colorimetry.pdf

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  To what extent is my understanding correct?


The biggest problem with your understanding is that the values in a LMS triplet do not correspond 1:1 with the # of photons collected in some sensor. They are only relative values. They indicate how the L cone was stimulated relative to the M and S cone, for example. 

Every other color space besides LMS also indicates the relative retinal response, just less directly. So LMS is not unique in its capabilities but it is certainly more suitable for chromatic adaptation algorithms.