Using the Lorain County GIS Viewer (linked below), selecting LBRS Centerlines to be visible within GIS Layers, I see: The southern road is not Rush Street (not even visible, so probably a service drive); The northern road is Rush Street but not connected across the railroad tracks. The LBRS data is government-produced by Ohio so maybe I think it may be okay to use as a source for data for OSM, what do you think?

Not as a source but strictly for corroboration: Bing Streetside shows neither as labeled streets.

I'm inclined to remove Rush Street name from the bottom street name but not add it to the top (Bing Streetside does not corroborate LBRS Centerlines data for the northern street).

https://experience.arcgis.com/experience/96c021db7a504796bf5dce62b3578cd6