Answer in progress:
From this question: Install older package version in Alpine
I see that each new version of Alpine updates its packages, tossing out the older versions to stay lean.
So we need to find the version of Alpine that corresponds to the date that Tesseract 3.04 was released, and use FROM Alpine:3.5
in Dockerfile.
Then add tesseract-ocr
will add the only version available in that Alpine version.
It seems that Alpine 3.5 or 3.4 should have Tesseract 3.04.
EDIT: I've run into a problem, which is that FROM Alpine:3.5
fails. Investigating.
EDIT 2: the earliest official version of Alpine available on Docker Hub is 3.12, which is why alpine:3.5
fails. So...
If anyone knows a better way, please shout it out!