It’s a rubbish bin. The one (that should) be under my computer desk is a waste paper basket.
McDonald’s replaced the word TRASH on their in-store bins with RUBBISH or WASTE (I think) a few years ago, after the local English Nazis made an outcry about it. We don’t use the word trash as a noun, but it’s common as a verb (eg. what rock stars do to hotel rooms).
What you call a dumpster we call a skip in the UK. But this has to be qualified. A skip is the sort of thing you can hire for a few days when you want to clear away a lot of rubbish (say when your moving house). The thing is left on your driveway for a few days, you fill it up, then it’s taken away. Average cost, getting on for £100. This is the sort of thing I mean :- skip
But you only need to buy the vowels! Guess the consonants and put the vowels on layaway. Worked for my grandfather - a poor immigrant who came to this country with nothing but a dream and the shirt on his back. Look at him now - full English vocabulary with nouns, pronouns, adverbs and everything.
“Dumpster” has become a generic word, but originally it referred only to the Dempster Dumpster®, a patented bin that a special truck would pick up and dump into the truck. Now, it means any kind of trash bin that can be picked up by a truck. They range from the little ones http://scottnath.com/past/03-10-unsung2.shtml that a truck dumps and puts back to these huge ones that must be winched onto a truck and taken to the landfill. http://www.crash.com/people/michael/LifeInNYC/dumpster.html
If the recptacle has food waste in it, or any refuse that decomposes, rots or smells, it’s Garbage. If it merely has paper and plastic waste, it’s Trash.
Small waste containers in bedrooms and bathrooms are either wastebaskets or trash cans.
Larger recepatacles are Garbage Cans, or The Garbage.