Right.
It doesnt matter that the DO you shadowed did an allopathic residency, and OMM isn't the only way that the DO differs from the MD.
Specifically let the DO you're working with know you want a letter for DO. They are saavy to what their ed was all about and what the difference is and what schools are considering, if they're precepting you likely this isn't their first time writing an LOR.
If they're MD, it's 50/50 if they have any clue about DO or can craft your letter that way. You'll do yourself a favor doing some research and providing them with an LOR guideline for DO so they can at least direct their comments towards what those schools look for.
DO schools pride themselves on their uniqueness from the allopaths, they don't just see themselves as a second choice backup plan to allopath schools, which is why they are putting applicants through the paces about this.
I was guilty of not knowing about this stuff as a pre-med until I went to apply. MD or DO, do yourselves a favor and spend like an hour with wikipedia at least learning some basic history. In this day and age it's ridiculous for any candidate or MD not to know more about DO.
And not in the same category by an means, while you're at it, read about what PAs, RNs, NPs, chiropractors, pharmacists, and naturopaths and homeopaths do.
I had questions interview day asking me if I had considered other healthcare paths, or what I thought about alternative medicine, etc, and this is just ****e you need to know no matter what school you apply to.