As a rule of thumb, a one-line dictionary definition isn’t going to carry nearly as much nuance as an encyclopedia article. But you have a point that the word can be flexible.
A well-constructed back road would very well meet the definition of a byway to an American, who would consider a “small road” to have fewer than four lanes. The only qualification for a byway here is that it evokes nostalgia for a bygone era. But all you have to do is peruse a gallery of English byways to get the sense that byways there are mainly about something much more primitive. From a legal standpoint, the following photo is apparently of a byway open to all traffic, probably because it’s built well enough that it can be used by vehicles as an exception, but still conceptually closer to a cycleway, footway, or bridleway:
Even so, it seems like a byway can also be thought of as a subclass of highway, bringing us right back to the starting point.
Agreed. Maybe this triangulation would at least help us come up with more intuitive presets or documentation.