I went to both John Lobb and Edward Green websites and their derby shoes include both types. Now, it is a possibility that there was such distinction before, but it now becomes obsolete. I don't entirely dismiss your opinion, just that I haven't heard of that before.
Cosigning this. There may be individual entities who use a (possibly archaic?) distinction between the two illustrated versions of derbies but most reference books, websites, vendors and sales assistants use the terms in their more generic sense and would include both, so any actual distinction has rightly or wrongly fallen by the wayside in common parlance. It's now gotten to the point where the term 'oxford' bleeds over to include open laced shoes too, and 'loafer' to include just about any slip on shoe.
It's a bit like the term 'button down' which outside of America means a button collar shirt, but in the ever more casual US it's often used to denote a shirt with 'buttons down the front' as opposed to a polo or t shirt.