Technically, all the impeachment and conviction does is to remove the person in question from office. Their removal from office leaves them open to subsequent criminal prosecution, which is not possible while in office. That is why Ford pardoned Nixon. Absent that pardon we would all have had to endure the spectacle of a former President being put on trial and probably found guilty. Ford chose to think being the first to resign and being tarnished in history was punishment enough.
Because all it does is remove them from office, it would not be redefining the law in any way.
However, you are partially correct in that it was the Senate's consideration in each case that the "crimes and misdemeanors" cited in the articles of impeachment were not sufficiently criminal (or maybe of concern/valid at all) to warrant removal from office. Johnson was retained in office by one vote; Clinton's margin was larger.
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