You are aware that parts of Ukraine was controlled by Austro-Hungary at that point? It makes perfect sense to say that some Ukrainians lived in that Empire while most lived in the Russian Empire.
You original quote:
“Review the forms filled by people from the area who immigrated to the US in the early 1900s. There was no such thing as Ukrainian at the time.”
This claim is absolutely wrong, and I have proven it as so. Plenty of folk checked off Ruthenian for their language, which was what Americans referred to as Ukrainian.
For the final death blow to your original claim. Here is a note from the 1930 census when Ukrainian was added:
“Ukrainian has been differentiated from Ruthenian although the distinction is more a matter of country of birth than of language.”
https://www2.census.gov/prod2/decennial/documents/16440598v2ch08.pdf
There you have it. Your 1939 map shows the post-independence situation. Ukrainian had become the name used for folk from Ukraine while Ruthenian remained in use for the small Ukrainian speaking population across the border.