I think I know of the flags you speak of but can you get me the unit designation from the OHS website?
I used to know a guy in Dayton that owns the 16th OVI regimental flag and it was embroidered. I seem to recall its eagle was different than what we typically know for regimental flags. This was the 90 day (or was it six months?) regiment who then came back home and sort of reformed.
In actuality, the original US unit colors regulations actually call for embroidered flags but when mass producing them it was simply impractical to do so and so painting was done instead. Additonally, embroidery and silk don't get along over time. It is my understanding that makers did mass produce blank flags that were then sent to flag painters for decoration as sub-contractors. The Confederate Richmond Depot did that for the four divisional issues of flags in 1863.
There were a few flag makers in Ohio - one in Cleveland and the two in Cincy that I have already mentioned. You could also go across the Midwest and find others like Gilbert & Hubbard in Chicago and the Fancy Flag Bazaar in Indianapolis (also called the Grand Flag Bazaar) and even Evansville, IN had a flag maker that did some Indiana colors.
With the passing of Howard Madaus the top experts in the country for Union flags are Steve Hill and Les Jensen. Thankfully I know them both and lean on them for those things I do not know. So if I cannot help with the flag in question I know who to turn to.
Let me know when you can.