I heard back from my contact in Wren, OR. Here's what he said:

This is a complicated problem, I see this all the time traveling all over the country.
Numbers vs. names, maps vs. signage can make navigating confusing. In this
case, while I don't have any pictures, I can say that I have never seen a sign
saying Halsey-Sweet Home Highway. All the signage and addresses on that road
are OR228. The designation Halsey-Sweet Home Highway #212 is an ODOT
internal designation. This document shows the cross reference for Oregon
highways.
https://www.oregon.gov/odot/Data/Documents/Routes-to-Highway-Cross-
Reference-Table.pdf

For Oregon, in general, outside of city limits, the signage used is the US or State Highway number, not the ODOT name. Inside of city limits it's up to the city to decide whether to use the ODOT name the highway number or some other name. There are, of course, several exceptions to this.