This article quotes the Department of Education (emphasis mine):
Nearly ten percent of the nation’s schools – 8652 of some 91,000 – already face the first level of sanctions under the federal Elementary and Secondary Education Act (ESEA). According to a July 1 press release from the U.S. Department of Education (DoE), the schools failed to make sufficient progress on state assessments. The 3.5 million students in those schools – all of which received ESEA Title I funds – are eligible to transfer to other schools.
Whether or not that particular definition of 'failing' (ie, calculating progress via state assessment testing) is universally agreed upon is, of course, up to debate and interpretation.
I attempted to find more recent statistics but, alas, it appears that 15 years later, the louder narrative is how standardized testing is a poor measurement, so I'm not sure there's going to be stats to directly compare against the aforementioned ones.
Does the government, or third party, provide a definition for what constitutes a failing public school in the United States, K-12?
Yes. But it varies from department to department, legislation to legislation, private entity to private entity. So while there are definitions of "failing school" there isn't a single, universally shared definition.