Difference between Li2YCY and LiYCY
This might be a bit of speculation, but:

seeing the lighter Li2YCY is tested against a higher voltage, it's probably also the more expensive; so you'd pick the cheaper one if the peak voltage doesn't matter to you, or if weight is no concern, or if the higher outer diameter is not a problem.
These are coaxial cables. Thus, the same properties as for RF cable are relevant if you are using it for changing signals (i.e. different geometries and dielectrics lead to different impedances and velocities,  so you'll have to design driver and receiver to match that impedance if you're doing anything over this cable where the highest relevant signal frequency (note: for square waves that can be often be the 13th harmonic!) has a wavelength that's smaller than 10 times the cabling length). So even if Li2YCY is cheaper, and in every aspect better than LiYCY, when you've tuned your system to the latter, you wouldn't just randomly switch to the former
Compatibility: You've got a factory with LiYCY all over the place. Every machine has grommets that fits the outer diameter perfectly; you've got stock of replacement parts, tools... for your cabling. Now someone shows up and tells you that you can switch to this better cable, although you've never had a problem with the old one. Would you?
Compliance: You build aircraft control systems. You went through 3 years of certification for your fly-by-wire system. This incorporates the exact type of cabling you use. Switching out a 50 cm piece of cable could easily cost you hundreds of thousands in re-certification work.