The ideal dipole antenna is roughly a half wave long.
A monopole is half of that, with the other half in the conductive surface below the monopole. The colinear antenna is a series of stacked and phased half wave dipoles with a quarter wave section at the top and bottom.
If your commercial whip is only a half wavelength long or less, then it is just a monopole whip, and not a colinear.
As to how the colinear works... consider it as an antenna array rather than a single antenna. Each antenna is separately fed, and it is phased so that the radiation emitted from the antenna is in phase with the radiation from the other antennas, causing constructive interference in the desired radiation direction.
There are several styles of colinear antennas. A common commercial version uses folded dipoles (that look like flat/oval loops) mounted along a vertical rod. The version you describe combines the phasing harness (the coax) with the antenna elements. The shield and center conductor are swapped at half wavelength intervals to create the correct phase between each section.
Either way, the effect of this is to take the toroidal radiation pattern of the dipole and flatten it, to give decreased gain at higher elevation angles and increased gain horizontally. Sometimes (typically for colinears on towers) the length of the phasing is adjusted to cause the flattened toroid to "droop" to make it better fit the curvature of the Earth.
One of the advantages of using an array of antennas is that the radiation pattern can be "steered" by adjusting the phase between elements which shifts where the constructive and destructive interference occurs.