Hi, I'm not sure if I have all the answers you need. 

I think some reasons for this are:

1. The term cage-free' is more understandable for a non-technical audience than deep litter, that as you say is more of a technical term for veterinarians. This is important for public communication and campaigns.
2. A campaign message is often benefited by keeping it simple, in this case asking for 'cage-free' and then as companies and governments adopt agree, we see more deeply defined standards developed and adopted, that do not confine birds to cages and also include other criteria like you are mentioning in deep litter and free-range systems.
3. USDA and European Union use the term 'cage-free' to describe their standards as do influential organisations in animal welfare advocacy like the RSPCA. As significant advocacy on this issue has happened in these regions, and a large proportion of EA's come from Europe and english speaking countries where this term is used, the term has probably been adopted as a default.
4. The terminology might vary by region? I'm not sure where you are based. For example, working in Europe, I have most often heard 'deep litter' used when working with German companies.

Overall, I would emphasise, that cage-free is not a standard that is intended to compete with any other definition of housing for laying hens, rather it is a minimum ask, the birds shall not be confined to cages, which allows for significant reduction in their pain.