OSM uses its own service, Nominatim, as tommorris has posted. It's OSM's own code using OSM data. The data and code are both free, but the osm.org servers have limited capacity; you can download the code and data and run it yourself if you like.
Mapquest Open runs an instance of this. MQ's servers are beefier so they have more liberal usage than osm.org's instance.
'Mainstream' Mapquest uses commercial data and therefore has pretty strict ToUs. Those are the ones you've quoted above: they don't apply to Mapquest Open.