What affects the sale price of a vehicle in GTA Online?
After a bit of investigation I can answer (at least part of) my own question.
First, it seems that my initial recollection of the price given by LSC was wrong: you can indeed check the price of a car, add a modification, drive out, drive back in again and check the newly updated price (obviously this is only possible if you have not sold a car in the last 48 mins -- otherwise you will be told that they are not accepting any vehicles at the moment).
Here is what I have figured out so far:
After a modification, the price added to the sale price is exactly 50% of the cost of the modification.
Modifying the car with multiple items from the same category does not increase the sale price by the sum of all those modifications. For example, if you already have 'Engine Tuning 2' and you then fit 'Engine Tuning 1', the latter will REPLACE the former and the sale price will DROP by 50% of the difference between cost of the two mods.
Most of the modifications I've tried so far DO add to the sale price of the car. However, there may be some that don't (as I have not extensively tried all of them). I will update if I find any that do not.