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.
There's not really a good metric of "OSM quality" yet.
The only things I can suggest are:
- issues: number reported, number closed, diversity of reporters and closers - on the basis that if lots ofpeople are reporting issues, that means "many eyes make all bugs shallow" might be working. If lots of people are closing them, then someone is actually paying some attention.
- tag diversity. Human mappers tend to actually use a fairly wide variety of tags for points of interest.
- some measure of how much the data that has been imported from, say, government public domain GIS datasets, has been modified.
In the U.S., there are various efforts to compare the OSM data to TIGER data from after Census started their big quality improvement project. They go a long ways towards showing where people are actually working on OSM (which has a rough correlation with quality).
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.