This might be okay.
Maybe there is no such an excellent translation. Consider that even in English, the term “unclassified” also causes much confusion and does not give the sense that it is a low-important road. So the iD Editor has to call it “Minor/Unclassified Road”.
While “ถนนชั้น…” can be confusing with layers when used with ordinary numbers, but it should not be when used with “เอก, โท, ตรี”, as these words already imply ranking by themselves. (I also avoided the confusion when proposing it in the first place.) The problem with “ชั้นเอก, ชั้นโท, ชั้นตรี” is that it’s an old-fasion language.
“ถนนสายย่อย” should be avoided for tertiary, IMO, as its implied relative size is unclear.
Let’s throw in some other random ideas for tertiary (if not "ชั้นเอก, ชั้นโท, ชั้นตรี): ถนนสาขา, ถนนกิ่ง?
After conversations with Steve Coast, he admits that the tagging scheme is imperfect, even for the UK, where it originated. I personally think we can settle on what we have now, it’s imperfect but mostly logical.