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I think they're typically separate here, but it's dump and recycling center in generalities in my area. My town specifically has a shared facility though and generally it's just called the drop off center but I imagine different areas call them different things.

We have separate wheeled bins for household waste, recycling, and compost (which includes garden waste but specifically bars thick branches or roots that I guess could mess up the trucks?). For things that don't fit in those bins you can either schedule a pick up specially or you can take it to the drop off center. Our drop off center does not charge extra for recyclables but they do charge an entrance fee (a couple dollars) and there are additional fees for things like extra household waste bags or things which require special disposal (like house paint or batteries or appliances like an iron or kettle etc). They actually have a matrix and diagrams on the website to say what they accept and what has additional charges and I think even how to organize your car so it's easy to drop things off in order of what comes first in the facility.