"Kosz" is an interesting word to me. Here it means basket - but I am sure that I learned this word when someone pointed to a small garbage!! So if it's a "waste basket" or small garbage bin - it could also be referred to as "kosz" or just a basket without saying it was for garbage (because it's obvious). :) I hope that made some sense! :) (so interesting)
Expect you would rather use 'koszyk' in most cases when referring to a basket. 'Kosz', for me, is by default associated with a garbage bin, unless the context is known. I'd definitely translate this sentence as 'into the garbage/waste/whatever bin'. Also usually you 'wrzucasz coś do kosza' = 'throw' instead of 'wkładasz'. But you're right, 'kosz' has quite a broad meaning.
I used the word placing instead of putting however it was not accepted... are they not synonyms that can be used interchangeable for 'wkłada'?
"My mother is putting the bread in the trash." This should be accepted. In my region, people say "the trash" and almost never "trash bin" or anything like that.
I don't think it's a trash bin in this sentence, because than we'd probably not use "wkłada", which implies putting something more or less carefully, but rather "wrzuca" - throwing inside, or "wyrzuca" which by itself implies throwing 'away'.
Well, 'trash bin' is at least "kosz" in a way, "the trash" is getting too far from the original sentence, it's an interpretation and not a translation.