Just one small remark while I ponder the deeper stuff:
> Honestly the terminology [name of monoidal preorder]-enriched set would make a lot more sense. Oh well.
We'll soon see that
* preorders are a special case of categories
* monoidal preorders are a special case of monoidal categories, and
* categories enriched over monoidal preorders are a special case of categories enriched over monoidal categories.
So, our use of the term "enriched categories" is looking forward to this realization. The things we're calling enriched categories now _are_ enriched categories - just a special case.