DBA vs. PhD Perception within Academia
While the experience that individual has gained throughout the course of their career is the best indicator of whether an individual has earned the respect of their peers, my impression is that the PhD title is generally more recognized with the public at large (basically those unfamiliar with how advanced degrees are awarded within academic institutions). This is purely a social construct that isn't germane to the field of advanced study.  

I see the issue as someone calling a chiropractor a "Doctor" even though there is no award of an MD, or not understanding that a D.O. and an M.D. are basically the same thing.  

Now, some universities in Europe do not traditionally award a "PhD" because they have a different three letter acronym for their doctorates. That does not mean that the individual is any less accomplished, its just a different system than the one in the US.

Perhaps one way to explain it to your students is to note what other fields have a similar award. In law, there are Juris Doctorates and PhDs; and in art there are MFAs and PhDs. The primary distinction appears to be what you have already recognized, that one is more about application than research.

To be a bit snarky, there are some fields of study altogether that do not get much appreciation, education being the main one with the award of the D.Ed. I do see some eye rolling when that one comes up, but I've also seen eye rolling from guys in the so called "pure" or "hard" sciences and their opinions of anyone in the humanities or "soft" sciences.

As to directly answer your question, Yes, there are published articles discussing the "perception" of one degree over another. I took a quick stroll on Google Scholar and found a number of published, peer-reviewed articles discussing this very question. 

Here is one titled, "Doctoral Differences: Professional doctorates and PhDs compared".