In Hebrew, words can have multiple layers of meaning. They can be used in one context and have one level of understanding but when used in a different context can mean something else idiomatically. (Just as "kick the bucket" does not mean to literally strike a pail with one's foot.)
While the word chet can mean to miss the mark, it can also be understood to miss the mark of appropriate behavior and can be seen to be the fault of the one who did the wrong behavior.
There are many words in Hebrew that refer to "sin." Each word has a slightly different connotation and rabbinic authorities stretching back centuries disagree as to exactly what word refers to what type of wrongdoing.
Some of the words used in rabbinic literature are: chet, pesha, avone, ashma, aveira.
Look at the Yom Kippur viduy (confession) for these words. The various commentaries on the prayers will give you different definitions of these words that all connote wrongdoing.