I think you explained it pretty well. "Universal income" is simpler language, but what happens is that people earning at a middle level pay themselves $12,000 (i.e. no change) and those earning more pay themselves 12K and pay some more to cover the 12K payments to those earning less.

Milton Friedman advocated a negative income tax along the lines you describe. Experiments found it was challenging to administer a NIT which

1. provides an income guarantee as generous as the cash and in-kind benefits already available to many welfare recipients in the United States,

2. provides an ostensible incentive to work (a far greater concern when benefits are to be extended beyond the traditional welfare population dominated by female-headed families), and

3. restricts coverage to any manageable proportion of the population—the so-called "break-even" problem.

The program eventually gave birth to the Earned Income Tax Credit which is still on the books.

a 1990 IRS study revealed that owing in part to the complexity of the EIC rules, almost 40 percent of EIC benefits were paid to families who were not eligible for them. Yet the finding of such a high error rate did not deter Congress from both enlarging and further complicating the EIC in the fall of 1990

UBI might one of those ideas that looks better on paper than it works in practice. But it might not have to be very good to be an improvement over what we have.