Garbage, trash and rubbish refer to the actual waste. Depending on which country you live in, you use one of the terms. In Australia, it is by far most common for people to say rubbish. Trash is a very American term. As for the container, the one in the house = rubbish bin. The one outside the house we also tend to call rubbish bin or wheelie bin, because in Australia they come with wheels. Garbage can and trash can are not used commonly in Australia, but everyone would know what you mean anyway. If you stick with rubbish/rubbish bin, everyone in Australia and the UK will understand you without a problem.

In the UK, we just use the word "bin" in every context - indoor and outdoor - and the meaning is clear. If you want to be more precise, then there are certain kinds of bins like "waste paper basket" (for dry rubbish, perhaps for your study or lounge), "wheelie bins" (outdoor bins with wheels), "litter bins" (outdoors, provided for public use). But be assured that "bin" is enough. Over to the Americans for their words!